Problem 49733. Solve the arithmetic differential equation D(n) = n

Created by ChrisR

Solve Later

{"parts":[{"partUri":"/matlab/document.xml","contentType":"application/vnd.mathworks.matlab.code.document+xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\"?><w:document xmlns:w=\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\"><w:body><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:hyperlink w:docLocation=\"https://www.mathworks.com/matlabcentral/cody/problems/47843-compute-the-arithmetic-derivative-of-integers\"><w:r><w:rPr><w:u/></w:rPr><w:t>Cody Problem 47843</w:t></w:r></w:hyperlink><w:r><w:t> involved the arithmetic derivative of integers. In particular, </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"D(p) = 1\"/></w:customXmlPr><w:r><w:t>D(p) = 1</w:t></w:r></w:customXml><w:r><w:t> if </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"p\"/></w:customXmlPr><w:r><w:t>p</w:t></w:r></w:customXml><w:r><w:t> is prime and </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"D(mn) = n D(m) + m D(n)\"/></w:customXmlPr><w:r><w:t>D(mn) = n D(m) + m D(n)</w:t></w:r></w:customXml><w:r><w:t>. Therefore, the arithmetic derivatives of 1, 2, 3, 4, 5, and 6 are 0, 1, 1, 4, 1, and 5, respectively. </w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>One might then ask about solving arithmetic differential equations (ADEs). Because the study of differential equations often starts with solving </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"dy/dx = y\"/></w:customXmlPr><w:r><w:t>dy/dx = y</w:t></w:r></w:customXml><w:r><w:t>, let’s consider the analogous ADE </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"D(n) = n\"/></w:customXmlPr><w:r><w:t>D(n) = n</w:t></w:r></w:customXml><w:r><w:t>. The definition of the arithmetic derivative shows that no prime can solve this equation, but the sample calculations above show that the first (i.e., </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"m = 1\"/></w:customXmlPr><w:r><w:t>m = 1</w:t></w:r></w:customXml><w:r><w:t>) solution is 4. </w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Write a function to compute the </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"m\"/></w:customXmlPr><w:r><w:t>m</w:t></w:r></w:customXml><w:r><w:t>th solution to this ADE. Because the solutions become large quickly, return the </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>logarithm</w:t></w:r><w:r><w:t> of the solution. </w:t></w:r></w:p></w:body></w:document>","relationship":null}],"relationships":[{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/document","target":"/matlab/document.xml","relationshipId":"rId1"}]}

<div 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; "><div style="block-size: 165px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 82.5px; transform-origin: 407px 82.5px; vertical-align: baseline; "><div 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: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><a target='_blank' href = "https://www.mathworks.com/matlabcentral/cody/problems/47843-compute-the-arithmetic-derivative-of-integers">Cody Problem 47843</a><span 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: 182.033px 7.79167px; transform-origin: 182.033px 7.79167px; unicode-bidi: normal; "> involved the arithmetic derivative of integers. In particular, <img src="data:image/png;base64,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" alt="D(p) = 1" style="width: 61.5px; height: 18.5px;" width="61.5" height="18.5"><span 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: 7.38333px 7.79167px; transform-origin: 7.38333px 7.79167px; unicode-bidi: normal; "> if p<span 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: 42.0083px 7.79167px; transform-origin: 42.0083px 7.79167px; unicode-bidi: normal; "> is prime and <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAUQAAAAlCAYAAAA6NZCNAAAIcUlEQVR4nO1dW5HjMBBsDmYQAiFgBEGwDMIgDEIhGAIhHEIhGEJh78Oecnuix8iRbGtPXeWq25Mf0qjnodHYARoaGhoaGhoaGhoaGhoaGhoaGhoa6sERQLd1JyI4bt2B/xD9Ss9pc7stcup/B+CQ6V6b4AfAbetOGPAD4I79G+6/ghsGma+B63g0rI/c+t9h0NPs3OkBXIzHUk9eizEUnAA88X8bxR/YOHHG8shrTWMouKIuLpZG7fr/RAEOdRg6/EvHC5MgruODfwG8x7+t4eppvFdtxuWMwQP9zzhgmnc5Hph4ccMwt8wXK67YTr53DPPbMKBm/e/G+2dPh/SYC8T1gBMGgYhgYh7jYDxvr3iiKc4FEyfehnOeiCuL8GgrJym8bDnFCTXrfxGjy6R+Bc47Yq4gocHeMEQUtaLHMMaqk7df4oFpvkPLnjPsRjE1miyBC4Z+lkA/HjWtimrX/wcy54etxAfm4bWPVIexvdboUPDC/51z4qghlqt50bk+cv5g2+hQIPwskcMMRVl7Re36/zM+L1vwwsQ/Rc5lL+EbtOSXascZ+1DgLaCXUTEZcJTok9meHMwdZSKYGg3iX9D/NzKtPE5IIz7U+a5o4I39EP8byOSvvRu6B1wRjwQY2oBqxdqbLMWA506J1GYQ/4r+35HJCDPxrR6TQ2x9jShGCvH78Xz2NgcMpL3gk1wdtfkI3Y33O2M+yUdMO2jWya/RuB8wjZ9xwlQyEwLvMFs9LyuKvkbyVCkGqAQv+N4lDHQpg3jEMHfc3w5TiZQrspO20NJ1D/ovY+MxxMamsYRfTiwhPgtERw/SsVj+QIqgXbtbkmviRK8Yr5Nq45rBbuzbS10rbfp5lsjnYTzPBWuNV+hIdSwPzOUjxvyIzzIaX66vU+dZc0GhewtnYijBi1B/cxdr5zSIPeYG6xeTIztiznN+pp7r0K76Vvp/VvdhI6rn0rKU743nBZGD+FogknS15t24xgkYhPLCYOmZDB21nRxtDF4GSCT4HK+Rf1uXCFZFduE3w7GkXo9l02NSHomWYw6B5WfNoeolszY0r8DzXCjBC42UqMiKEhEij+mIaT4lqmKjIW0XR5vG3vSf57JX/Y85LuHfV3lE2Z1h4sWghaiXk6kGhO8jwhBhinCfGAbsanPlDXhcx7FPvESUPlqUQci4ZGMlR4S4xOOJbIRkQjBBLCLgnUSrQda5KB3ZphqfErzQeCJ/+U0Jgyh8lSJknk/m+iHSprEH/Zco947PuWQnGzPW0q+vIv4cxNe5qBSB8I7VFZ8FluI9ZNeKSfaCe0KAuVJc8Dm2mEFgWJcAewLLRjsDC8l4KWYtTtdLO20QUgxiKV5oLIn+Y681cr9D56VAlo93TE5AIHJ/juNhRxRzDlvrv5TnyH204xZ5WlcpX0f8S4ivX/PRnidFILLT98bnRLMnemE+0SxIVwTFr5RpZWJls3jx2gwiy+aFT6KzzGPXp0Q6rrwtI4WspXihscQg6rzXksMajQFzvsqrkwxxDq6SJpkTX9S0tf5zhCopHYbkkq3G+iuDqIlv2Z3pME92ujqaIhBOnutJY0+kB8mC1J4jVifFXseCbLtXKyFWDyjz4yOZJqkFOn/oUq4UspbghQuyAZcCa4R4C5yT8kqofguIwc5Bz3XMOexN/13cSDXW1pWBEyxoKyl4wn07VzJIC0Iekz2Rfg7nkDR4XC7hPAJtLnyzqbL2LjMwJ5nr2lCbvj5VRiEuvWE3iCV44XvO3jdVWLb6nuwAtNGIOYc96D+/G63vxU7WYqy/3lQJeWEX5GXtmNUWocUIoXeCNTjZ6mtzXccE0oJkj2o1NEuiCIFruZR6pO4yh0jOMveRjOfYsuzUO9K+1IKUA6XcLycvXPgqogjcM5dBZL66DDdvnvnafAZ/a/1ng+eaA8mNpq5SFteVphBfl6qElFQIHbsne3qtnBzO6wFyW0//J/0M9ZGVTbxmzPu8PfeyYO1dZh6fi7CcgAcGGfjyq77IgnGETUn42bF7luCFr++xPi9BToPIUZ5L0UMbSNo5aFlsrf+6lEhDnifGOqanEvEuknvKe6r6u3gx79sZz5MJc3kwTqrrvv2oNinkBeYGwUUg7TVjH68UJavlM2BMMheBtJJISYuAl0SxZWeP+ZInpgAyN7HNqRK8cEHOz/1GSU6DqMunGOy8tEy1c5BCbVc5y1b6z5tBrmdy/+WFi5BMb0jbrJqBFce3pd1hGJSQXgpBLZDyAB94Ml3GJlQnyFFOj/lES5tvTDKWK6bathBKve9aCkIyl9xY5icMBNLOgInvW2od8PnFE6t85AOjPpTihQvZ3n1VyGkQudxGI1SSwoGB1OG6SnW20n9eybkMJ+c3xSDHlsKL35/Wrz+JwHmZpt9VTF2Xi/f1KYoeMCMmLJ3A58kMRRfc7rrWhQfq+XI2e1WX3PSuIo9LyP+rDq6l019OlkLaFEjdoA+leKEhu6UlIv9cBpGdg0v/Qs5Pb37wPO1B/zkV4OIQt1tWaBLxJss8VjLAO5vf1t29EP4unm8H9UD9cAlUPs7gulau8/U9dK2rH5Yl3l4QkxswfdjBVYpkzWd+o+giU1+kUYoXrueU+qxbLoPYI1zEHeK6ODj9YZO96H9sbID9Ix1AJR+jlhxTLctNjTvq/MrN3nFBua9VWyCvMi7ejYygR31fzC6BtfRfIukq7MyWPyb0DWr9caxasOXv1dTKyRqxhqyr++2j2n7lTF6Qr+UDnzWi2K+lRdB+YnZ9lNT/Kn9WVn7msNQSJScs2/sNeXDAurKW3dZmDNdFKf2PlcztHhfsn4xnVJKL+COQ5P8aqIF/fxk55S9fTW9oaGhoaGhoaGhoaGhoaGhoaGhoaGhoaGjIjX98m2HjhsI+6wAAAABJRU5ErkJggg==" alt="D(mn) = n D(m) + m D(n)" style="width: 162px; height: 18.5px;" width="162" height="18.5"><span 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: 296.208px 7.79167px; transform-origin: 296.208px 7.79167px; unicode-bidi: normal; ">. Therefore, the arithmetic derivatives of 1, 2, 3, 4, 5, and 6 are 0, 1, 1, 4, 1, and 5, respectively. </div><div 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: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span 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: 382.108px 7.79167px; transform-origin: 382.108px 7.79167px; unicode-bidi: normal; ">One might then ask about solving arithmetic differential equations (ADEs). Because the study of differential equations often starts with solving <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAIIAAAAlCAYAAACOGjGrAAADvUlEQVR4nO2bfZGDMBDFnwcc1AAGUFAFOKiDOqgFNFRCPdQCGrDQ+yN5x4aGfJCkLUN+M525KRwJ7zabtwsHVCqVSqWya04Amm9PovJ9HgCu355Exc3pA9d/ZRqnQc0s2WgAdAB6qFX6KDzeBcAz4fc7AGfMcy0duIfhCuAOtUpfAG6FxxuhgmELPdT8ONeUgKpY6DCLey44Tos828KngvZw9FDCTii7596gsk8K9BgvqACuZGSAEra0Pxihgi6FTwXtIZmgxC1Z0p2R549HP1M6aA8H9+3SqXbQn1Q+EbSHoIVy7Vf98xVzqiUsKVmm2cwdj7UBYzb6+rFG9IS5rD3DNLW2ceW9Len093vwFbn1N+ig0inLt7P+eYQS9i7OY43ucuc8NsKf7nt9XignqOwx6bE7qFLxifegBeYSmMdfMEvUAWbJGbI9UYfUT2yFVEL/fy6wi8BssBSOcDK2vT3GtN0RXup1+poTzGiX2WCt8mDmkR6Cop7174VuKVKblE/0ihXk0h/A3ICxRY682bX0w+M2tx/SiWS5FyIIx5vwnsJlILgqD5rJEXPW22JQc2WEFHOcQ38Ac9SsGUGmTFfa5gqzGb0B/i5haEu5hdsIyntxpduLOG9E2or8BVL1N9LkWloePcc5mC2jNPo73x74DJks5jS4FjSD5ziRVVCOKuXbpOpvGCTbybJD53Lza1nlCv++H9pSlmOspf2QoCW+BbAnUvS3mqYl9A4+syEDhis7NBpDW8ouUwSYe6Wv/JNbTEylsuRbVcOSFP0NY2Xbb0MCRbIsMW8IW22hLWXOdW0uvkAhDcwyM+UB1y9UDWSr/sZN2NK+fOzMQHEJJvdnrjifG+Yq9p0ng9Z2c9L8MVDWXkgZoO5N+oStzzZ+oWogW/QH4A4ENndkqm2gRF6LXrlPPREmbmhL2RUILczVzaAdLHNYNq2WbpvX2iNb9Adgrgg2kRoooUeYL3dQINfF5fVCH/bEtJT5R5NNpIueqwzqHnOAtXougz5ngpnVZD+BQbLXUnKL/v/I9C8v0sBMtxPC9/Flt2+NHu9tYBdyPsseQLv4nllDBoitAdV7ju+NGP3f4IOJ5cMJBkOP8D1MulYfMS1lwgdgfMAk6fXYcsXzHlzOvPcc3xMx+hfjgvA3ixrkc8sVRYz+xegQ16/n3l7JQ6z+RWgRb7JCW8oVP1v0zwKriQdmpx1jsnL+88oRSdU/G7LS2BKJN9R3CVNI1T8bsprYQo63lI9Mqv4/wx31FfNKpVKpVH6JPxg13ByqmkTvAAAAAElFTkSuQmCC" alt="dy/dx = y" style="width: 65px; height: 18.5px;" width="65" height="18.5"><span 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: 108.783px 7.79167px; transform-origin: 108.783px 7.79167px; unicode-bidi: normal; ">, let’s consider the analogous ADE <img src="data:image/png;base64,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" alt="D(n) = n" style="width: 60px; height: 18.5px;" width="60" height="18.5"><span 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: 153.25px 7.79167px; transform-origin: 153.25px 7.79167px; unicode-bidi: normal; ">. The definition of the arithmetic derivative shows that no prime can solve this equation, but the sample calculations above show that the first (i.e., <img src="data:image/png;base64,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" alt="m = 1" style="width: 40px; height: 18px;" width="40" height="18"><span 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: 45.1167px 7.79167px; transform-origin: 45.1167px 7.79167px; unicode-bidi: normal; ">) solution is 4. </div><div 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: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span 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: 98.6583px 7.79167px; transform-origin: 98.6583px 7.79167px; unicode-bidi: normal; ">Write a function to compute the m<span 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: 244.142px 7.79167px; transform-origin: 244.142px 7.79167px; unicode-bidi: normal; ">th solution to this ADE. Because the solutions become large quickly, return the <span 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: 28.7917px 7.79167px; transform-origin: 28.7917px 7.79167px; unicode-bidi: normal; ">logarithm<span 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 7.79167px; transform-origin: 1.94167px 7.79167px; unicode-bidi: normal; "> of the solution. </div></div></div>

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Problem 49733. Solve the arithmetic differential equation D(n) = n

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