{"group":{"id":1,"name":"Community","lockable":false,"created_at":"2012-01-18T18:02:15.000Z","updated_at":"2025-12-14T01:33:56.000Z","description":"Problems submitted by members of the MATLAB Central community.","is_default":true,"created_by":161519,"badge_id":null,"featured":false,"trending":false,"solution_count_in_trending_period":0,"trending_last_calculated":"2025-12-14T00:00:00.000Z","image_id":null,"published":true,"community_created":false,"status_id":2,"is_default_group_for_player":false,"deleted_by":null,"deleted_at":null,"restored_by":null,"restored_at":null,"description_opc":null,"description_html":null,"published_at":null},"problems":[{"id":60940,"title":"Find the first occurence of a given gap between two consecutive prime numbers","description":"Problem statement \r\n\r\nGiven a gap = p' - p between the two consecutive prime numbers p and p', find its first occurence, f.\r\n\r\nExamples\r\n\r\nIf , f=2, since 5 - 3 = 2, and 3 is the 2nd prime;\r\nIf , f=4, since 11 - 7 = 4, and 7 is the 4th prime;\r\nIf , f=9, since 29 - 23 = 6, and 23 is the 9th prime;\r\nIf  neither equals an even positive integer nor equals 1 your function should return the empty set : f = []; \r\n\r\nSee also\r\nProblem 60939. Frequencies of prime gaps\r\nPrime numbers properties II","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 403.3px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 408px 201.65px; transform-origin: 408px 201.65px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; 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display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 9.10833px 8px; transform-origin: 9.10833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ep', \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 75.4417px 8px; transform-origin: 75.4417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003efind its first occurence, f.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 32.675px 8px; transform-origin: 32.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eExamples\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; 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height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 5.825px 8px; transform-origin: 5.825px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, f\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 7.98333px 8px; transform-origin: 7.98333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e=4\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 93.4083px 8px; transform-origin: 93.4083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e, since 11 - 7 = 4, and 7 is the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 10.5px 8px; transform-origin: 10.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e4th\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.3917px 8px; transform-origin: 21.3917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e prime;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIf\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"40\" height=\"18\" style=\"width: 40px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 5.825px 8px; transform-origin: 5.825px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, f\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 7.98333px 8px; transform-origin: 7.98333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e=9\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 101.708px 8px; transform-origin: 101.708px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e, since 29 - 23 = 6, and 23 is the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 10.5px 8px; transform-origin: 10.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e9th\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.3917px 8px; transform-origin: 21.3917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e prime;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 5.825px 8px; transform-origin: 5.825px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIf \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: normal; font-weight: 400; color: rgb(0, 0, 0);\"\u003eΔ\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 158.333px 8px; transform-origin: 158.333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e neither equals an even positive integer nor equals \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 5.83333px 8px; transform-origin: 5.83333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e1 \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 132.625px 8px; transform-origin: 132.625px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eyour function should return the empty set : \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 14.975px 8px; transform-origin: 14.975px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003ef = []\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e; \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28.3917px 8px; transform-origin: 28.3917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eSee also\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/problems/60939-frequencies-of-prime-gaps\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eProblem 60939. Frequencies of prime gaps\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/groups/95759\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ePrime numbers properties II\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function f = first_occurence_of_prime_gap(delta)\r\n  f = delta;\r\nend","test_suite":"%%\r\ndelta = 2;\r\nf_correct = 2;\r\nassert(isequal(first_occurence_of_prime_gap(delta),f_correct))\r\n\r\n%%\r\ndelta = 4;\r\nf_correct = 4;\r\nassert(isequal(first_occurence_of_prime_gap(delta),f_correct))\r\n\r\n%%\r\ndelta = 6;\r\nf_correct = 9;\r\nassert(isequal(first_occurence_of_prime_gap(delta),f_correct))\r\n\r\n%%\r\ndelta = 8;\r\nf_correct = 24;\r\nassert(isequal(first_occurence_of_prime_gap(delta),f_correct))\r\n\r\n%%\r\ndelta = 14;\r\nf_correct = 30;\r\nassert(isequal(first_occurence_of_prime_gap(delta),f_correct))\r\n\r\n%%\r\ndelta = 10;\r\nf_correct = 34;\r\nassert(isequal(first_occurence_of_prime_gap(delta),f_correct))\r\n\r\n%%\r\ndelta = 12;\r\nf_correct = 46;\r\nassert(isequal(first_occurence_of_prime_gap(delta),f_correct))\r\n\r\n%%\r\ndelta = 1;\r\nf_correct = 1;\r\nassert(isequal(first_occurence_of_prime_gap(delta),f_correct))\r\n\r\n%%\r\ndelta = 3;\r\nf_correct = [];\r\nassert(isequal(first_occurence_of_prime_gap(delta),f_correct))\r\n\r\n%%\r\ndelta = -1 +1i*pi;\r\nf_correct = [];\r\nassert(isequal(first_occurence_of_prime_gap(delta),f_correct))\r\n\r\n%% Forbidden functions\r\nfiletext = fileread('first_occurence_of_prime_gap.m');\r\nillegal = contains(filetext, 'regexp') || contains(filetext, 'str2num') || contains(filetext, 'assignin') || contains(filetext, 'echo')\r\nassert(~illegal);","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":149128,"edited_by":149128,"edited_at":"2025-07-26T06:47:33.000Z","deleted_by":null,"deleted_at":null,"solvers_count":25,"test_suite_updated_at":"2025-07-09T05:56:06.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2025-06-25T12:22:19.000Z","updated_at":"2026-03-30T01:19:31.000Z","published_at":"2025-06-25T13:06:05.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem statement \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven a gap \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\Delta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e= p' - p\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e between the two consecutive prime numbers \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep', \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003efind its first occurence, f.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExamples\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\Delta = 2\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, f\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e=2\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, since 5 - 3 = 2, and 3 is the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e2nd\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e prime;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\Delta = 4\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, f\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e=4\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, since 11 - 7 = 4, and 7 is the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e4th\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e prime;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\Delta = 6\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, f\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e=9\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, since 29 - 23 = 6, and 23 is the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e9th\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e prime;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\Delta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e neither equals an even positive integer nor equals \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e1 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eyour function should return the empty set : \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef = []\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e; \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eSee also\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/cody/problems/60939-frequencies-of-prime-gaps\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eProblem 60939. Frequencies of prime gaps\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/cody/groups/95759\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ePrime numbers properties II\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":60953,"title":"Chek the Delta =  p' - p = 6k gap theorem about arithmetic progressions in the prime number set","description":"Context\r\n \r\nIn the prime numbers set there are some arithmetic progressions (sequences of three or more consecutive prime numbers (p, p’, p’’) equally spaced one to the others by an even number  ).\r\n \r\nOne theorem, which can actually easily be proven from , is that above the sole and unique triplet (3, 5, 7) -with a gap of  then-  all the following progressions are such that \r\n \r\nProblem statement\r\n\r\nFor a given interval [i1, i2], i1 \u003e 7 and i2 \u003e 7 find p the first corresponding arithmetic progression (3 or more consecutive consecutive primes equally spaced) in this interval and check the conjecture equation simply by calculating the integer ratio . \r\n\r\nExamples\r\n                \r\nFor [i1, i2] = [8, 68], p = [47, 53, 59] and k = [1, 1], since this is the first arithmetic progression above 8 and with  here; \r\n \r\nFor [i1, i2] = [180, 228], p = [199, 211, 223], and k = [2, 2], since this is the first arithmetic progression above 180 and with  here;\r\n\r\nFor [i1, i2] = [240, 272], p = [251, 257, 263, 269], and k = [1, 1, 1], since this is the first arithmetic progression above 140 and with  here; \r\n\r\nFor [i1, i2] = [180, 272], p = [199, 211, 223], and k = [2, 2], since this is the first arithmetic progression above 180 and with  here;\r\n\r\nTip\r\n \r\nFirst maybe, train yourself to find the first and last indices of zeros of a first block of consecutive zeros in a vector of integers, eg for u = [1 0 0 0 1 1 0 0 1], j1 = 2 and j2 = 4.\r\n\r\nForbidden functions\r\n \r\n \r\nregexp\r\n \r\nstr2num\r\n \r\nassignin\r\n\r\necho\r\n \r\n \r\nSee also\r\n \r\nProblem 60940. Find the first occurence of a given gap between two consecutive prime numbers\r\nPrime numbers properties I","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 1469.2px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 408px 734.6px; transform-origin: 408px 734.6px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 26.0583px 8px; transform-origin: 26.0583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eContext\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 21px; text-align: left; transform-origin: 385px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 248.55px 8px; transform-origin: 248.55px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIn the prime numbers set there are some arithmetic progressions (sequences of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 110.45px 8px; transform-origin: 110.45px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003ethree or more consecutive prime numbers\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 30.825px 8px; transform-origin: 30.825px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e(p, p’, p’’)\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 166.875px 8px; transform-origin: 166.875px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e equally spaced one to the others by an even number \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"134.5\" height=\"18\" style=\"width: 134.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 6.21667px 8px; transform-origin: 6.21667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e ).\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 21px; text-align: left; transform-origin: 385px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 172.317px 8px; transform-origin: 172.317px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eOne theorem, which can actually easily be proven from \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"236.5\" height=\"19\" style=\"width: 236.5px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 85.5667px 8px; transform-origin: 85.5667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, is that above the sole and unique triplet \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 24.1083px 8px; transform-origin: 24.1083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e(3, 5, 7)\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 45.8917px 8px; transform-origin: 45.8917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e -with a gap of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"40\" height=\"18\" style=\"width: 40px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 17.8917px 8px; transform-origin: 17.8917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e then-\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 134.208px 8px; transform-origin: 134.208px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eall the following progressions are such that \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"97\" height=\"19\" style=\"width: 97px; height: 19px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.0083px 8px; transform-origin: 63.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem statement\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 31.5px; text-align: left; transform-origin: 385px 31.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 61.4583px 8px; transform-origin: 61.4583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor a given interval \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 75.825px 8px; transform-origin: 75.825px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e[i1, i2], i1 \u0026gt; 7 and i2 \u0026gt; 7\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.1667px 8px; transform-origin: 15.1667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e find \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 4.275px 8px; transform-origin: 4.275px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ep\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 217.05px 8px; transform-origin: 217.05px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e the first corresponding arithmetic progression (3 or more consecutive consecutive primes equally spaced) in this interval and check the conjecture equation simply by calculating the integer ratio \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"57\" height=\"18.5\" style=\"width: 57px; height: 18.5px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 32.675px 8px; transform-origin: 32.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eExamples\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 31.0667px 8px; transform-origin: 31.0667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e                \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 40.8667px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 20.4333px; transform-origin: 392px 20.4333px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 20.4333px; text-align: left; transform-origin: 364px 20.4333px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 12.4417px 8px; transform-origin: 12.4417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 48.4167px 8px; transform-origin: 48.4167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e[i1, i2] = [8, 68]\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 48.0333px 8px; transform-origin: 48.0333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ep = [47, 53, 59]\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 28.2px 8px; transform-origin: 28.2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ek = [1, 1]\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 158.692px 8px; transform-origin: 158.692px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, since this is the first arithmetic progression above \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e8\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 29.95px 8px; transform-origin: 29.95px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and with \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"90\" height=\"18\" style=\"width: 90px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 19.8333px 8px; transform-origin: 19.8333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e here; \u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 40.8667px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 20.4333px; transform-origin: 392px 20.4333px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 20.4333px; text-align: left; transform-origin: 364px 20.4333px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 12.4417px 8px; transform-origin: 12.4417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 60.0917px 8px; transform-origin: 60.0917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e[i1, i2] = [180, 228]\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 59.1917px 8px; transform-origin: 59.1917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ep = [199, 211, 223]\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 17.5px 8px; transform-origin: 17.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 32.0833px 8px; transform-origin: 32.0833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ek = [2, 2], \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 154.808px 8px; transform-origin: 154.808px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003esince this is the first arithmetic progression above \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 11.675px 8px; transform-origin: 11.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e180\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and with \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"97.5\" height=\"18\" style=\"width: 97.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 17.8917px 8px; transform-origin: 17.8917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e here;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 40.8667px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 20.4333px; transform-origin: 392px 20.4333px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 20.4333px; text-align: left; transform-origin: 364px 20.4333px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 12.4417px 8px; transform-origin: 12.4417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 60.0917px 8px; transform-origin: 60.0917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e[i1, i2] = [240, 272]\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 75.2667px 8px; transform-origin: 75.2667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ep = [251, 257, 263, 269]\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 17.5px 8px; transform-origin: 17.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 39.8583px 8px; transform-origin: 39.8583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ek = [1, 1, 1], \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 154.808px 8px; transform-origin: 154.808px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003esince this is the first arithmetic progression above \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 11.675px 8px; transform-origin: 11.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e140\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 29.95px 8px; transform-origin: 29.95px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and with \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"90\" height=\"18\" style=\"width: 90px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 19.8333px 8px; transform-origin: 19.8333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e here; \u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 40.8667px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 20.4333px; transform-origin: 392px 20.4333px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 20.4333px; text-align: left; transform-origin: 364px 20.4333px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 12.4417px 8px; transform-origin: 12.4417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 60.0917px 8px; transform-origin: 60.0917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e[i1, i2] = [180, 272]\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 59.1917px 8px; transform-origin: 59.1917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ep = [199, 211, 223]\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 17.5px 8px; transform-origin: 17.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 32.0833px 8px; transform-origin: 32.0833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ek = [2, 2], \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 154.808px 8px; transform-origin: 154.808px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003esince this is the first arithmetic progression above \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 11.675px 8px; transform-origin: 11.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e180\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and with \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"97.5\" height=\"18\" style=\"width: 97.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 17.8917px 8px; transform-origin: 17.8917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e here;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 10.3667px 8px; transform-origin: 10.3667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eTip\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 21px; text-align: left; transform-origin: 385px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 358.192px 8px; transform-origin: 358.192px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFirst maybe, train yourself to find the first and last indices of zeros of a first block of consecutive zeros in a vector of integers, eg for \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 122.325px 8px; transform-origin: 122.325px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eu = [1 0 0 0 1 1 0 0 1], j1 = 2 and j2 = 4.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 67.6417px 8px; transform-origin: 67.6417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eForbidden functions\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 20.4333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 10.2167px; transform-origin: 392px 10.2167px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.4px 8px; transform-origin: 21.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eregexp\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 20.4333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 10.2167px; transform-origin: 392px 10.2167px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.2833px 8px; transform-origin: 25.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003estr2num\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 20.4333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 10.2167px; transform-origin: 392px 10.2167px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.6833px 8px; transform-origin: 25.6833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eassignin\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 20.4333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 10.2167px; transform-origin: 392px 10.2167px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15.175px 8px; transform-origin: 15.175px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eecho\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28.3917px 8px; transform-origin: 28.3917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eSee also\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/problems/60940-find-the-first-occurence-of-a-given-gap-between-two-consecutive-prime-numbers\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eProblem 60940. Find the first occurence of a given gap between two consecutive prime numbers\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/groups/95630\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ePrime numbers properties I\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [p, k] = check_the_6k_delta_theorem(i1, i2)\r\n  \r\n    u = i1;\r\n    k = i2;\r\n\r\nend","test_suite":"%%\r\ni1 = 8;\r\ni2 = 68;\r\np_correct = [47, 53, 59];\r\nk_correct = [1, 1];\r\n[p,k] = check_the_6k_delta_theorem(i1,i2);\r\nassert(isequal(p,p_correct) \u0026 isequal(k,k_correct))\r\n\r\n\r\n%%\r\ni1 = 180;\r\ni2 = 228;\r\np_correct = [199, 211, 223];\r\nk_correct = [2, 2];\r\n[p,k] = check_the_6k_delta_theorem(i1,i2);\r\nassert(isequal(p,p_correct) \u0026 isequal(k,k_correct))\r\n\r\n\r\n%%\r\ni1 = 240;\r\ni2 = 272;\r\np_correct = [251, 257, 263, 269];\r\nk_correct = [1, 1, 1];\r\n[p,k] = check_the_6k_delta_theorem(i1,i2);\r\nassert(isequal(p,p_correct) \u0026 isequal(k,k_correct))\r\n\r\n\r\n%%\r\ni1 = 180;\r\ni2 = 272;\r\np_correct = [199, 211, 223];\r\nk_correct = [2, 2];\r\n[p,k] = check_the_6k_delta_theorem(i1,i2);\r\nassert(isequal(p,p_correct) \u0026 isequal(k,k_correct))\r\n\r\n\r\n%% Test forbidden functions\r\nfiletext = fileread('check_the_6k_delta_theorem.m');\r\nillegal = contains(filetext, 'regexp') || contains(filetext, 'str2num') || contains(filetext, 'assignin') || contains(filetext, 'echo')\r\nassert(~illegal);","published":true,"deleted":false,"likes_count":4,"comments_count":3,"created_by":149128,"edited_by":149128,"edited_at":"2025-07-26T07:04:42.000Z","deleted_by":null,"deleted_at":null,"solvers_count":34,"test_suite_updated_at":"2025-07-11T05:36:42.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2025-07-05T06:40:29.000Z","updated_at":"2026-03-06T14:30:46.000Z","published_at":"2025-07-10T12:03:38.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eContext\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn the prime numbers set there are some arithmetic progressions (sequences of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ethree or more consecutive prime numbers\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e(p, p’, p’’)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e equally spaced one to the others by an even number \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\mathbf{\\\\Delta = p' - p = p'' - p'}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e ).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eOne theorem, which can actually easily be proven from \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\forall p \\\\in \\\\mathbb{P}, p\u0026gt; 3 \\\\Rightarrow \\\\exists n \\\\in \\\\mathbb{N}^*, p = 6n \\\\pm 1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, is that above the sole and unique triplet \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e(3, 5, 7)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e -with a gap of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\Delta = 2\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e then-\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eall the following progressions are such that \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\mathbf{\\\\Delta = 6k}, k \\\\in \\\\mathbb{N}^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem statement\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor a given interval \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e[i1, i2], i1 \u0026gt; 7 and i2 \u0026gt; 7\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e find \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e the first corresponding arithmetic progression (3 or more consecutive consecutive primes equally spaced) in this interval and check the conjecture equation simply by calculating the integer ratio \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\mathbf{k = \\\\Delta / 6}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExamples\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e                \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e[i1, i2] = [8, 68]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep = [47, 53, 59]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ek = [1, 1]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, since this is the first arithmetic progression above \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e8\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and with \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\Delta = 6 = \\\\mathbf{1} \\\\times 6\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e here; \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e[i1, i2] = [180, 228]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep = [199, 211, 223]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ek = [2, 2], \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003esince this is the first arithmetic progression above \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e180\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and with \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\Delta = 12 = \\\\mathbf{2} \\\\times 6\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e here;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e[i1, i2] = [240, 272]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep = [251, 257, 263, 269]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ek = [1, 1, 1], \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003esince this is the first arithmetic progression above \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e140\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and with \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\Delta = 6 = \\\\mathbf{1} \\\\times 6\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e here; \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e[i1, i2] = [180, 272]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep = [199, 211, 223]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ek = [2, 2], \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003esince this is the first arithmetic progression above \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e180\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and with \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\Delta = 12 = \\\\mathbf{2} \\\\times 6\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e here;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eTip\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFirst maybe, train yourself to find the first and last indices of zeros of a first block of consecutive zeros in a vector of integers, eg for \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eu = [1 0 0 0 1 1 0 0 1], j1 = 2 and j2 = 4.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eForbidden functions\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eregexp\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003estr2num\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eassignin\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eecho\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eSee also\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/cody/problems/60940-find-the-first-occurence-of-a-given-gap-between-two-consecutive-prime-numbers\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eProblem 60940. Find the first occurence of a given gap between two consecutive prime numbers\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/cody/groups/95630\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ePrime numbers properties I\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"}],"problem_search":{"errors":[],"problems":[{"id":60940,"title":"Find the first occurence of a given gap between two consecutive prime numbers","description":"Problem statement \r\n\r\nGiven a gap = p' - p between the two consecutive prime numbers p and p', find its first occurence, f.\r\n\r\nExamples\r\n\r\nIf , f=2, since 5 - 3 = 2, and 3 is the 2nd prime;\r\nIf , f=4, since 11 - 7 = 4, and 7 is the 4th prime;\r\nIf , f=9, since 29 - 23 = 6, and 23 is the 9th prime;\r\nIf  neither equals an even positive integer nor equals 1 your function should return the empty set : f = []; \r\n\r\nSee also\r\nProblem 60939. Frequencies of prime gaps\r\nPrime numbers properties II","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 403.3px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 408px 201.65px; transform-origin: 408px 201.65px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 64.95px 8px; transform-origin: 64.95px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem statement \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 39.675px 8px; transform-origin: 39.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eGiven a gap \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: normal; font-weight: 400; color: rgb(0, 0, 0);\"\u003eΔ\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21.3667px 8px; transform-origin: 21.3667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e= p' - p\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 142.367px 8px; transform-origin: 142.367px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e between the two consecutive prime numbers \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ep\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 9.10833px 8px; transform-origin: 9.10833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003ep', \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 75.4417px 8px; transform-origin: 75.4417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003efind its first occurence, f.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 32.675px 8px; transform-origin: 32.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eExamples\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 61.3px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 30.65px; transform-origin: 392px 30.65px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 5.825px 8px; transform-origin: 5.825px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIf \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"40\" height=\"18\" style=\"width: 40px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 5.825px 8px; transform-origin: 5.825px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, f\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 7.98333px 8px; transform-origin: 7.98333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e=2\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 90.0333px 8px; transform-origin: 90.0333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e, since 5 - 3 = 2, and 3 is the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 12.4417px 8px; transform-origin: 12.4417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e2nd\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.3917px 8px; transform-origin: 21.3917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e prime;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 5.825px 8px; transform-origin: 5.825px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIf \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"40\" height=\"18\" style=\"width: 40px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 5.825px 8px; transform-origin: 5.825px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, f\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 7.98333px 8px; transform-origin: 7.98333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e=4\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 93.4083px 8px; transform-origin: 93.4083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e, since 11 - 7 = 4, and 7 is the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 10.5px 8px; transform-origin: 10.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e4th\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.3917px 8px; transform-origin: 21.3917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e prime;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIf\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"40\" height=\"18\" style=\"width: 40px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 5.825px 8px; transform-origin: 5.825px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, f\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 7.98333px 8px; transform-origin: 7.98333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e=9\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 101.708px 8px; transform-origin: 101.708px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e, since 29 - 23 = 6, and 23 is the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 10.5px 8px; transform-origin: 10.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e9th\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.3917px 8px; transform-origin: 21.3917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e prime;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 5.825px 8px; transform-origin: 5.825px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIf \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: normal; font-weight: 400; color: rgb(0, 0, 0);\"\u003eΔ\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 158.333px 8px; transform-origin: 158.333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e neither equals an even positive integer nor equals \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 5.83333px 8px; transform-origin: 5.83333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e1 \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 132.625px 8px; transform-origin: 132.625px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eyour function should return the empty set : \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 14.975px 8px; transform-origin: 14.975px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003ef = []\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e; \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28.3917px 8px; transform-origin: 28.3917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eSee also\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/problems/60939-frequencies-of-prime-gaps\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eProblem 60939. Frequencies of prime gaps\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/groups/95759\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ePrime numbers properties II\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function f = first_occurence_of_prime_gap(delta)\r\n  f = delta;\r\nend","test_suite":"%%\r\ndelta = 2;\r\nf_correct = 2;\r\nassert(isequal(first_occurence_of_prime_gap(delta),f_correct))\r\n\r\n%%\r\ndelta = 4;\r\nf_correct = 4;\r\nassert(isequal(first_occurence_of_prime_gap(delta),f_correct))\r\n\r\n%%\r\ndelta = 6;\r\nf_correct = 9;\r\nassert(isequal(first_occurence_of_prime_gap(delta),f_correct))\r\n\r\n%%\r\ndelta = 8;\r\nf_correct = 24;\r\nassert(isequal(first_occurence_of_prime_gap(delta),f_correct))\r\n\r\n%%\r\ndelta = 14;\r\nf_correct = 30;\r\nassert(isequal(first_occurence_of_prime_gap(delta),f_correct))\r\n\r\n%%\r\ndelta = 10;\r\nf_correct = 34;\r\nassert(isequal(first_occurence_of_prime_gap(delta),f_correct))\r\n\r\n%%\r\ndelta = 12;\r\nf_correct = 46;\r\nassert(isequal(first_occurence_of_prime_gap(delta),f_correct))\r\n\r\n%%\r\ndelta = 1;\r\nf_correct = 1;\r\nassert(isequal(first_occurence_of_prime_gap(delta),f_correct))\r\n\r\n%%\r\ndelta = 3;\r\nf_correct = [];\r\nassert(isequal(first_occurence_of_prime_gap(delta),f_correct))\r\n\r\n%%\r\ndelta = -1 +1i*pi;\r\nf_correct = [];\r\nassert(isequal(first_occurence_of_prime_gap(delta),f_correct))\r\n\r\n%% Forbidden functions\r\nfiletext = fileread('first_occurence_of_prime_gap.m');\r\nillegal = contains(filetext, 'regexp') || contains(filetext, 'str2num') || contains(filetext, 'assignin') || contains(filetext, 'echo')\r\nassert(~illegal);","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":149128,"edited_by":149128,"edited_at":"2025-07-26T06:47:33.000Z","deleted_by":null,"deleted_at":null,"solvers_count":25,"test_suite_updated_at":"2025-07-09T05:56:06.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2025-06-25T12:22:19.000Z","updated_at":"2026-03-30T01:19:31.000Z","published_at":"2025-06-25T13:06:05.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem statement \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven a gap \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\Delta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e= p' - p\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e between the two consecutive prime numbers \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep', \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003efind its first occurence, f.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExamples\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\Delta = 2\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, f\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e=2\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, since 5 - 3 = 2, and 3 is the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e2nd\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e prime;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\Delta = 4\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, f\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e=4\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, since 11 - 7 = 4, and 7 is the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e4th\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e prime;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\Delta = 6\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, f\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e=9\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, since 29 - 23 = 6, and 23 is the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e9th\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e prime;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\Delta\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e neither equals an even positive integer nor equals \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e1 \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eyour function should return the empty set : \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ef = []\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e; \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eSee also\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/cody/problems/60939-frequencies-of-prime-gaps\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eProblem 60939. Frequencies of prime gaps\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:hyperlink w:docLocation=\\\"https://fr.mathworks.com/matlabcentral/cody/groups/95759\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ePrime numbers properties II\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":60953,"title":"Chek the Delta =  p' - p = 6k gap theorem about arithmetic progressions in the prime number set","description":"Context\r\n \r\nIn the prime numbers set there are some arithmetic progressions (sequences of three or more consecutive prime numbers (p, p’, p’’) equally spaced one to the others by an even number  ).\r\n \r\nOne theorem, which can actually easily be proven from , is that above the sole and unique triplet (3, 5, 7) -with a gap of  then-  all the following progressions are such that \r\n \r\nProblem statement\r\n\r\nFor a given interval [i1, i2], i1 \u003e 7 and i2 \u003e 7 find p the first corresponding arithmetic progression (3 or more consecutive consecutive primes equally spaced) in this interval and check the conjecture equation simply by calculating the integer ratio . \r\n\r\nExamples\r\n                \r\nFor [i1, i2] = [8, 68], p = [47, 53, 59] and k = [1, 1], since this is the first arithmetic progression above 8 and with  here; \r\n \r\nFor [i1, i2] = [180, 228], p = [199, 211, 223], and k = [2, 2], since this is the first arithmetic progression above 180 and with  here;\r\n\r\nFor [i1, i2] = [240, 272], p = [251, 257, 263, 269], and k = [1, 1, 1], since this is the first arithmetic progression above 140 and with  here; \r\n\r\nFor [i1, i2] = [180, 272], p = [199, 211, 223], and k = [2, 2], since this is the first arithmetic progression above 180 and with  here;\r\n\r\nTip\r\n \r\nFirst maybe, train yourself to find the first and last indices of zeros of a first block of consecutive zeros in a vector of integers, eg for u = [1 0 0 0 1 1 0 0 1], j1 = 2 and j2 = 4.\r\n\r\nForbidden functions\r\n \r\n \r\nregexp\r\n \r\nstr2num\r\n \r\nassignin\r\n\r\necho\r\n \r\n \r\nSee also\r\n \r\nProblem 60940. Find the first occurence of a given gap between two consecutive prime numbers\r\nPrime numbers properties I","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 1469.2px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 408px 734.6px; transform-origin: 408px 734.6px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 26.0583px 8px; transform-origin: 26.0583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eContext\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 21px; text-align: left; transform-origin: 385px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 248.55px 8px; transform-origin: 248.55px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIn the prime numbers set there are some arithmetic progressions (sequences of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 110.45px 8px; transform-origin: 110.45px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003ethree or more consecutive prime numbers\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 30.825px 8px; transform-origin: 30.825px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e(p, p’, p’’)\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 166.875px 8px; transform-origin: 166.875px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e equally spaced one to the others by an even number \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg 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width=\"134.5\" height=\"18\" style=\"width: 134.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 6.21667px 8px; transform-origin: 6.21667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e ).\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 21px; text-align: left; transform-origin: 385px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 172.317px 8px; transform-origin: 172.317px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eOne theorem, which can actually easily be proven from \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"236.5\" height=\"19\" style=\"width: 236.5px; height: 19px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 85.5667px 8px; transform-origin: 85.5667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, is that above the sole and unique triplet \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 24.1083px 8px; transform-origin: 24.1083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e(3, 5, 7)\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 45.8917px 8px; transform-origin: 45.8917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e -with a gap of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAFAAAAAkCAYAAAAeor16AAAC70lEQVRoQ+1YPUgeQRDVXgmaSttYKKSwiQohbQRBLAIaxMIuSZFSiBYhpDCQlCmSCHYi0c4mEBELRclPk0JIiliYQisDwfTJe7ID67l7397Owbef7MHj7vtuZ2727dvZ2W1vy5eKgXaVdTZuywQqRZAJzAQqGVCaZwVmApUMKM1jFXiM774GXii/n4p5NwJ5AowD/SaoE9x3gUfAb1+gMQTehbOPwF+gMxUGFHHcgu0G0OPxwX4OAoeu9zEEfoKjYePsPu5riuBTMP2OIK4BK8CWCege7tNAh/ntFUtVAm/A4U+r1z/wPJACC5ExzMPuMTABfC344LQ+sJTJqfy2+J2qBFJtY8aJjE4rq5C5/LmLGNPHh7i/Mc/ruE9pCOSInAJLwB9gzjj7jPtIpAKaacbZxEWityQI6TObbAKjGgJfwpgyZkLlZU/lIfwuToFmklPnt/8ZZxTOg1gCORJHwL41CpzOk8ahc3Tq7EWTfNkKdKaq0BzIXPAK4OpEsnhx+f9idawPz86lPrDzVLikhUATZ7M6ZwNz3nuANaFzqocSyGTLvFdcce2SxinxCkykSKDMMu9CGUKgjILLibwTnq7jwVu1VyAzhaZSspWmpxACqTIWmr56j+qUKl6rwhSIkxjYb14s26K3cpLnFuDEt++1a6Wrsr1jOpkBXAX2hUFupEDmgDvAzQZT8wzvpbAuIzslhflioSCehpBHB2UESg4ImZb2AhCrwhQWkUrkNSLwHRpwQ+09ibCG0K6X+HfM9q7ZBIaQx35eyIc+BUrhvAqDS9W3R/t2Yd1qhwxypDWLvkmdW+wmOdkDbtsk+gjkKcUiwALyV2Di4koth5GxKgz8VK3NSN428AFY9njuwv/PgJ2ioHwE2qVJbLStcMgg5MkCWNZX5nZ7J3be1kWgXZbEkid2dW6rtLEU7blIfgNCyKOtMy01KmPqDvrK+csEKoc0E5gJVDKgNM8KzAQqGVCaZwVmApUMKM2zApUE/gdVDoAl3uI4wAAAAABJRU5ErkJggg==\" width=\"40\" height=\"18\" style=\"width: 40px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 17.8917px 8px; transform-origin: 17.8917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e then-\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 134.208px 8px; transform-origin: 134.208px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eall the following progressions are such that \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"97\" height=\"19\" style=\"width: 97px; height: 19px;\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.0083px 8px; transform-origin: 63.0083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eProblem statement\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 31.5px; text-align: left; transform-origin: 385px 31.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 61.4583px 8px; transform-origin: 61.4583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor a given interval \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 75.825px 8px; transform-origin: 75.825px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e[i1, i2], i1 \u0026gt; 7 and i2 \u0026gt; 7\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.1667px 8px; transform-origin: 15.1667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e find \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 4.275px 8px; transform-origin: 4.275px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ep\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 217.05px 8px; transform-origin: 217.05px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e the first corresponding arithmetic progression (3 or more consecutive consecutive primes equally spaced) in this interval and check the conjecture equation simply by calculating the integer ratio \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"57\" height=\"18.5\" style=\"width: 57px; height: 18.5px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 32.675px 8px; transform-origin: 32.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eExamples\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 31.0667px 8px; transform-origin: 31.0667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e                \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 40.8667px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 20.4333px; transform-origin: 392px 20.4333px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 20.4333px; text-align: left; transform-origin: 364px 20.4333px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 12.4417px 8px; transform-origin: 12.4417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 48.4167px 8px; transform-origin: 48.4167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e[i1, i2] = [8, 68]\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 48.0333px 8px; transform-origin: 48.0333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ep = [47, 53, 59]\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 28.2px 8px; transform-origin: 28.2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ek = [1, 1]\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 158.692px 8px; transform-origin: 158.692px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, since this is the first arithmetic progression above \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.89167px 8px; transform-origin: 3.89167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e8\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 29.95px 8px; transform-origin: 29.95px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and with \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"90\" height=\"18\" style=\"width: 90px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 19.8333px 8px; transform-origin: 19.8333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e here; \u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 40.8667px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 20.4333px; transform-origin: 392px 20.4333px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 20.4333px; text-align: left; transform-origin: 364px 20.4333px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 12.4417px 8px; transform-origin: 12.4417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 60.0917px 8px; transform-origin: 60.0917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e[i1, i2] = [180, 228]\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 59.1917px 8px; transform-origin: 59.1917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ep = [199, 211, 223]\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 17.5px 8px; transform-origin: 17.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 32.0833px 8px; transform-origin: 32.0833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ek = [2, 2], \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 154.808px 8px; transform-origin: 154.808px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003esince this is the first arithmetic progression above \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 11.675px 8px; transform-origin: 11.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e180\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and with \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"97.5\" height=\"18\" style=\"width: 97.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 17.8917px 8px; transform-origin: 17.8917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e here;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 40.8667px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 20.4333px; transform-origin: 392px 20.4333px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 20.4333px; text-align: left; transform-origin: 364px 20.4333px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 12.4417px 8px; transform-origin: 12.4417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 60.0917px 8px; transform-origin: 60.0917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e[i1, i2] = [240, 272]\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 75.2667px 8px; transform-origin: 75.2667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ep = [251, 257, 263, 269]\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 17.5px 8px; transform-origin: 17.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 39.8583px 8px; transform-origin: 39.8583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ek = [1, 1, 1], \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 154.808px 8px; transform-origin: 154.808px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003esince this is the first arithmetic progression above \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 11.675px 8px; transform-origin: 11.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e140\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 29.95px 8px; transform-origin: 29.95px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and with \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"90\" height=\"18\" style=\"width: 90px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 19.8333px 8px; transform-origin: 19.8333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e here; \u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 40.8667px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 20.4333px; transform-origin: 392px 20.4333px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 20.4333px; text-align: left; transform-origin: 364px 20.4333px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 12.4417px 8px; transform-origin: 12.4417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 60.0917px 8px; transform-origin: 60.0917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003e[i1, i2] = [180, 272]\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 59.1917px 8px; transform-origin: 59.1917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ep = [199, 211, 223]\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 17.5px 8px; transform-origin: 17.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 32.0833px 8px; transform-origin: 32.0833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; font-weight: 700; \"\u003ek = [2, 2], \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 154.808px 8px; transform-origin: 154.808px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003esince this is the first arithmetic progression above \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 11.675px 8px; transform-origin: 11.675px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e180\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and with \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"97.5\" height=\"18\" style=\"width: 97.5px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 17.8917px 8px; transform-origin: 17.8917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e here;\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 10.3667px 8px; transform-origin: 10.3667px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eTip\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 21px; text-align: left; transform-origin: 385px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 358.192px 8px; transform-origin: 358.192px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFirst maybe, train yourself to find the first and last indices of zeros of a first block of consecutive zeros in a vector of integers, eg for \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 122.325px 8px; transform-origin: 122.325px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eu = [1 0 0 0 1 1 0 0 1], j1 = 2 and j2 = 4.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 67.6417px 8px; transform-origin: 67.6417px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eForbidden functions\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 20.4333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 10.2167px; transform-origin: 392px 10.2167px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 21.4px 8px; transform-origin: 21.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eregexp\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 20.4333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 10.2167px; transform-origin: 392px 10.2167px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.2833px 8px; transform-origin: 25.2833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003estr2num\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 20.4333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 10.2167px; transform-origin: 392px 10.2167px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 25.6833px 8px; transform-origin: 25.6833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eassignin\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 8px; transform-origin: 0px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cul style=\"block-size: 20.4333px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 392px 10.2167px; transform-origin: 392px 10.2167px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 364px 10.2167px; text-align: left; transform-origin: 364px 10.2167px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 15.175px 8px; transform-origin: 15.175px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-style: italic; \"\u003eecho\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ul\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28.3917px 8px; transform-origin: 28.3917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eSee also\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.94167px 8px; transform-origin: 1.94167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/problems/60940-find-the-first-occurence-of-a-given-gap-between-two-consecutive-prime-numbers\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eProblem 60940. Find the first occurence of a given gap between two consecutive prime numbers\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 385px 10.5px; text-align: left; transform-origin: 385px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003ca target='_blank' href = \"https://fr.mathworks.com/matlabcentral/cody/groups/95630\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ePrime numbers properties I\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [p, k] = check_the_6k_delta_theorem(i1, i2)\r\n  \r\n    u = i1;\r\n    k = i2;\r\n\r\nend","test_suite":"%%\r\ni1 = 8;\r\ni2 = 68;\r\np_correct = [47, 53, 59];\r\nk_correct = [1, 1];\r\n[p,k] = check_the_6k_delta_theorem(i1,i2);\r\nassert(isequal(p,p_correct) \u0026 isequal(k,k_correct))\r\n\r\n\r\n%%\r\ni1 = 180;\r\ni2 = 228;\r\np_correct = [199, 211, 223];\r\nk_correct = [2, 2];\r\n[p,k] = check_the_6k_delta_theorem(i1,i2);\r\nassert(isequal(p,p_correct) \u0026 isequal(k,k_correct))\r\n\r\n\r\n%%\r\ni1 = 240;\r\ni2 = 272;\r\np_correct = [251, 257, 263, 269];\r\nk_correct = [1, 1, 1];\r\n[p,k] = check_the_6k_delta_theorem(i1,i2);\r\nassert(isequal(p,p_correct) \u0026 isequal(k,k_correct))\r\n\r\n\r\n%%\r\ni1 = 180;\r\ni2 = 272;\r\np_correct = [199, 211, 223];\r\nk_correct = [2, 2];\r\n[p,k] = check_the_6k_delta_theorem(i1,i2);\r\nassert(isequal(p,p_correct) \u0026 isequal(k,k_correct))\r\n\r\n\r\n%% Test forbidden functions\r\nfiletext = fileread('check_the_6k_delta_theorem.m');\r\nillegal = contains(filetext, 'regexp') || contains(filetext, 'str2num') || contains(filetext, 'assignin') || contains(filetext, 'echo')\r\nassert(~illegal);","published":true,"deleted":false,"likes_count":4,"comments_count":3,"created_by":149128,"edited_by":149128,"edited_at":"2025-07-26T07:04:42.000Z","deleted_by":null,"deleted_at":null,"solvers_count":34,"test_suite_updated_at":"2025-07-11T05:36:42.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2025-07-05T06:40:29.000Z","updated_at":"2026-03-06T14:30:46.000Z","published_at":"2025-07-10T12:03:38.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eContext\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn the prime numbers set there are some arithmetic progressions (sequences of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ethree or more consecutive prime numbers\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e(p, p’, p’’)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e equally spaced one to the others by an even number \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\mathbf{\\\\Delta = p' - p = p'' - p'}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e ).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eOne theorem, which can actually easily be proven from \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\forall p \\\\in \\\\mathbb{P}, p\u0026gt; 3 \\\\Rightarrow \\\\exists n \\\\in \\\\mathbb{N}^*, p = 6n \\\\pm 1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, is that above the sole and unique triplet \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e(3, 5, 7)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e -with a gap of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\Delta = 2\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e then-\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eall the following progressions are such that \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\mathbf{\\\\Delta = 6k}, k \\\\in \\\\mathbb{N}^*\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eProblem statement\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor a given interval \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e[i1, i2], i1 \u0026gt; 7 and i2 \u0026gt; 7\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e find \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e the first corresponding arithmetic progression (3 or more consecutive consecutive primes equally spaced) in this interval and check the conjecture equation simply by calculating the integer ratio \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\mathbf{k = \\\\Delta / 6}\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExamples\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e                \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e[i1, i2] = [8, 68]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep = [47, 53, 59]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ek = [1, 1]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, since this is the first arithmetic progression above \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e8\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and with \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\Delta = 6 = \\\\mathbf{1} \\\\times 6\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e here; \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e[i1, i2] = [180, 228]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep = [199, 211, 223]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ek = [2, 2], \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003esince this is the first arithmetic progression above \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e180\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and with \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\Delta = 12 = \\\\mathbf{2} \\\\times 6\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e here;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e[i1, i2] = [240, 272]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep = [251, 257, 263, 269]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ek = [1, 1, 1], \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003esince this is the first arithmetic progression above \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e140\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and with \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\Delta = 6 = \\\\mathbf{1} \\\\times 6\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e here; \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e[i1, i2] = [180, 272]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ep = [199, 211, 223]\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ek = [2, 2], \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003esince this is the first arithmetic progression above \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e180\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and with \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\Delta = 12 = \\\\mathbf{2} \\\\times 6\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e here;\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eTip\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFirst maybe, train yourself to find the first and last indices of zeros of a first block of consecutive zeros in a vector of integers, eg for \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eu = [1 0 0 0 1 1 0 0 1], j1 = 2 and j2 = 4.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc 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