Problem 55595. Easy Sequences 87: Perfect Power Modular Residue of a Nested Sum-product Function

Created by Ramon Villamangca

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:r><w:t>For a positive integer </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>x</w:t></w:r></w:customXml><w:r><w:t>, we define the function </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>Y</w:t></w:r></w:customXml><w:r><w:t>, as follows:</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> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>Y(1)=1</w:t></w:r></w:customXml><w:r><w:t>; and</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> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>Y(x)=1+1\\cdot(2+2\\cdot(3+3\\cdot(4\\dots(x-2)+(x-2)\\cdot((x-1)+(x-1)\\cdot x))))))</w:t></w:r></w:customXml><w:r><w:t> for </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>x&gt;1</w:t></w:r></w:customXml><w:r><w:t>.</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>Hence, for </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>x=6</w:t></w:r></w:customXml><w:r><w:t> we have:</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> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/></w:customXmlPr><w:r><w:t>Y(6)=1+1\\cdot(2+2\\cdot(3+3\\cdot(4+4\\cdot(5+5\\cdot6))))=873</w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>And if </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>x=10</w:t></w:r></w:customXml><w:r><w:t>:</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> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/></w:customXmlPr><w:r><w:t>Y(10)=1+1\\cdot(2+2\\cdot(3+3\\cdot(4+4\\cdot(5+5\\cdot(6+6\\cdot(7+7\\cdot(8+8\\cdot(9+9\\cdot10))))))))=4037913</w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>-----------------------------</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:rPr><w:b/></w:rPr><w:t>We now consider the following congruence:</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:rPr><w:b/></w:rPr><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>Y(x) \\equiv a^N\\ (mod\\ M)</w:t></w:r></w:customXml><w:r><w:rPr><w:b/></w:rPr><w:t> with </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>M</w:t></w:r></w:customXml><w:r><w:rPr><w:b/></w:rPr><w:t>and </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>N&gt;1</w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>The congruence expresses the possibility that the nested sum and product function defined above (</w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>Y</w:t></w:r></w:customXml><w:r><w:t>), can have a perfect power residue in some modular base. In fact, solving for </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>x</w:t></w:r></w:customXml><w:r><w:t>, the congruence always have a trivial solution, namely </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>x=1</w:t></w:r></w:customXml><w:r><w:t>, that's because </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>Y(1)=1</w:t></w:r></w:customXml><w:r><w:t>, which is, of course, a perfect power in any modular base. </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:rPr><w:b/></w:rPr><w:t>Given the value of integers </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>N</w:t></w:r></w:customXml><w:r><w:rPr><w:b/></w:rPr><w:t>, </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>M</w:t></w:r></w:customXml><w:r><w:rPr><w:b/></w:rPr><w:t>, and certain limit </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>L</w:t></w:r></w:customXml><w:r><w:rPr><w:b/></w:rPr><w:t>, find the sum </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>S</w:t></w:r></w:customXml><w:r><w:rPr><w:b/></w:rPr><w:t> of all positive integer values of </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>x\\le L</w:t></w:r></w:customXml><w:r><w:rPr><w:b/></w:rPr><w:t>, that satisfies the above congruence.</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>For </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>N = 3</w:t></w:r></w:customXml><w:r><w:t>, </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>M=7</w:t></w:r></w:customXml><w:r><w:t> and </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>L=20</w:t></w:r></w:customXml><w:r><w:t>, we see that only </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>x=1</w:t></w:r></w:customXml><w:r><w:t>and </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>x=5</w:t></w:r></w:customXml><w:r><w:t> satisfies the congruence, since: </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> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>Y(1)=1\\equiv 1^3\\ (mod\\ 7)</w:t></w:r></w:customXml><w:r><w:t>; and </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> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>Y(5)=1+1\\cdot(2+2\\cdot(3+3\\cdot(4+4\\cdot5))) = 153 \\equiv 153+7\\cdot121 \\equiv 1000 \\equiv 10^3 \\equiv 3^3\\ (mod\\ 7)</w:t></w:r></w:customXml><w:r><w:t>.</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>Therefore in this case </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>S(20,7,3) = 1 + 5 = 6</w:t></w:r><w:r><w:t>.</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>For </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>N = 3</w:t></w:r></w:customXml><w:r><w:t>, </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>M=10</w:t></w:r></w:customXml><w:r><w:t> and </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>L=20</w:t></w:r></w:customXml><w:r><w:t>, the above equation is satisfied, </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>\\forall x\\in \\{1,2,3 \\cdots20\\}</w:t></w:r></w:customXml><w:r><w:t>.</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>Therefore: </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>S(20,10,3) = sum(1:20) = 210</w:t></w:r><w:r><w:t>.</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: 20.440001px; 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: none; white-space: normal; "><div style="block-size: 594px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 297px; transform-origin: 407px 297px; vertical-align: baseline; "><div 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: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; 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; 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" width="502" height="19" style="width: 502px; height: 19px;"><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: 0px 0px; transform-origin: 0px 0px; "> for <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEkAAAAkCAYAAADFGRdYAAAAAXNSR0IArs4c6QAAAERlWElmTU0AKgAAAAgAAYdpAAQAAAABAAAAGgAAAAAAA6ABAAMAAAABAAEAAKACAAQAAAABAAAASaADAAQAAAABAAAAJAAAAABLVRfgAAAC6ElEQVRoBe2YO2hUQRSGE1ETRXyBSRGDbwUrK8UiwSoxaISggoKComh8IFqKGJM04qsQQS0khSikECxSiSAWgpVIsLDS+BYVXyi+X98ve2BYYryzzmx2cX749s7MnTvnzH9n753dioqk5EByIDmQHEgOlIEDI8hxHWwug1yLnqLMWQO34Secg6ScA5UcV8ItkDlGJpPk7P+gFia5Hg5BX6lPuJ4Ea4c5yVXEL+mVtJEEB+A41MFw6L1v0GJ/3XQHx8AuuAOnYDqUtTShVuiAfTAOXDVTOQIr3MYhytWc2wmPwJb8V8o9MAeKoaUEsdiZHtxDJaWH3Dv47gx61LngmNOuJTzWOfe3YhUddsBDsIS/UT4P8yGmgpo0l0y1cqaCTNBktL+Q9oDatALuwmUoRDJrO7hm/aB+ARZADAU1yU3wIhW7402UP4G+aqE0moG2wQOwODrqdb0QQiqaSVo5bvJ6DsWQzGqH++DGu0S9AUIomkmLyM6S1gR8nj+FTExmbYF7YHF1vApL4F/kbVLWLcBNstJXTLoBH36X4n18YeheOJsXopH66ry2kqlq5bwA3c2nkbOawPj74RXYKtKbT6/rEG8+75VE3Ew6SS9LWMfZma7y6yRzDsBrsFhaUWdgFoRSFJOWk532Sq5R+nkRShMZqBNccz5SPwH1EFrBTaolw2egjeUk0B5Gd7kHpFFwGiar4imN1wVvwFaONq+HQXFjqYWBLV5BO+6RDLAM9BOhGq7BdVC71A8KoOeS2mTQFaiErJI53fAWLFmtIrUVYjaXeWkDvS2uNq3e2sQVNoDuqh7W05xRDjrnn1B+mXfe6TposY1W15zn1PfC+EF7h2+cyZD9YHPU/Ob5htHr1QbQ22Vx3gAzqH/O9dHd993gdeSufcxxN+itWQzpbxn99LHHhc1RR7UNwFbILG0cm+BPE6jhnP4ZmAK+WssF7VDle2HqnxxIDiQHkgPJgeRAciA5kBwoSwd+ARw4yr706APYAAAAAElFTkSuQmCC" width="36.5" height="18" style="width: 36.5px; height: 18px;"><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: 0px 0px; transform-origin: 0px 0px; ">.</div><div 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: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; 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: 0px 0px; transform-origin: 0px 0px; ">Hence, for <img src="data:image/png;base64,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" width="36.5" height="18" style="width: 36.5px; height: 18px;"><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: 0px 0px; transform-origin: 0px 0px; "> we have:</div><div 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: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; 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: 0px 0px; transform-origin: 0px 0px; "> <img src="data:image/png;base64,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" 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RyZywD3A/zyHlrMr2ccjEmeTZj+upHBg8UsuTnJ4ECvsnAzELL/diVUctFpxlWnMCz2096rGjZ6p6S+v5Ssd2e8xwUDF/YbAy8AnSS9YewEYDBwUgJcGhXfWOCm/QcAJ/dxjVoarrIYfXCNhRmg4S7gDQsdyyHtEuBES15MSbH64BoLB4NMrpVE1sANxyCumx5MrvfieiQQq8Tsg2ss5HHLv+LoDnBOsG1O8t6NJa9NH1xjgVwdAcxJSBuI6wMA+/4Y4DjgaWAI8CQQq8Tog2sbmHuw1XIInmnksZ1iEdP+PjCKe0ubxGp/2tYY9q8++13R3S7tSMaz6LvupX30RTcGWzLcbyX5Z6i1H3Ag8J6DBWYfnJWjP9vIyzs/oVpvYAIwGDgpAS5ekjenmeNADOOYl7/dHGn4o6FHMq4EuoCjgL8DRSKNVNdhw8qGAXkdh2pmvthlvD7v9rHkLq9R5ymnbsbjmR2jKrZPlauP7TAg/aRq/10J5m+auHBeO65UrvWLSQkv4sp+3qSUbQf+suR04CpgCWB54BJAysNt1MJxmhvM25MrjeUm5VjeNCzCmUss9IRt5JmHJ19t2M5Q1cXqg2s7nAUiOOYsDfQHDgEmA6Z8Dw/8iBarxOqDaxswdk0ZgYcbAX4I4C9oKBsA5wPn8iFCidkH13bIo3WfJJNr2w/zFCPOa9MHnza4AxweAPDXoJTlAPb9w5LrOrj+A4hZYvTBtQ2eNYjlv1+3qPFs3po/VKjrnygw63O9N+3nO7tmvBir/WlzY9i/+ux3n4QDHwBcVy+Wdsby7FM2X/fRj8mWtOttnQNwHjgYOB5wWafTbtc+6HJ+EnK/kjenmeNAm+NYKX/5kov80VDi6fWWwE+AsYCLrJooveyiXEKHg4DIbLnJuJoDMg9DskRsFduz9GzpfZDIDX5V9LAVrmmNMyB9QPpEWQO48eVk9SBwbtlCWnyPJ/v7JfXz4OedBm3hAlEmCJ92GIn3ngNOAMx4GoLnvwFLATELOecC4zqAX5BEOMZwDP6aJDR09YmFU2DTusBBwGsN2Re6mhh9KBMLH4GYKcCFwFoAF2ay+cXtIqcBrusB6rchsfngEgvk9LMGWY/jfnVgX4AL5M8BRwPczFB4v9vcu3j+E7MPZWIhzeyKSNg2Sbwindkhz2374BILJpXkeZiZkNzvgetWlvQYk2LzwbUN+BFSZCBu0gfrzOO6Yw3eJDJFbiK4cg93k2EH13jdjWe55dgqEpP9YpNcY9i/vpwYI31IbLNdOQ+/CnBekDW5TU/SfMrmOz76Mdki/sq1D26qngHwfXPfImVnXdl+/JBxA3BOlpIl3bUPFp2fhNyvFM1pMYxjpf11ObVmO/0VeA+QTnAL7r8FuMoqieJM1xc89d429LkYyxPTZ9uALe/6DADyjlzPwM3l8lDhem+Fd/XV+RnYGI9fB2YA3FBL++K2UEL0336o5X8AHlgdCjCeOk1GweC+wGUJfO1fAi8cA3Bx/RtgAuAqrJe/QuIm1fz6UvT+GCjw48EKADdY3wXkwI0T1XcAtkus8hYMI1/LAgMA9uHtAJHRuPkd8JIkOFybiAVyfRxwJsD5oxOlTh+qtEHZWJA2mIObHwBcSP08SVwT1y2A25Pn2C8hfKgyHpEfl3mB4465zmD8pmP1PKRtBBwGUDgmXT33Lo7/1O1Dm7FAhr8EcF3INcGtQCdKVR+aiAWTV25ahicJXL+zfspKwJ+A/wQuAGKWOnyoEgsu4xH55HxMDOEDhOsHHvjfBiwKrANwfpB1Em7nfqjkNRY5HYZwfqa9mwG/BUYAbwLLATzENdd1PvNa1VhA1V4Sw/5V9kNcE7sI9Xk4Qv0pBS+UKZtFdqItJhVNnwMwFhgHPJAcnlxxcRL2wWUTTZaTJZwnRbrLjXENuV8pmtNiGMdC+mvQ+K9b/pyZDSrY5V9ZTnf8KT/fHemkPb/Sncm7fH/A/FnznnY3dCbPS7XfMF/84ACdJfwqR727sxQ6IN2Fuzw3NkUmB84scKIjRx/m6PDda4CyUtUH1tsLeAGQdv8lEz2kSv9lNRzMxgPkaR/AV/huVhswneXSN7ZHnh4PccoKN4fvAXcBslD2LetreEHa4H3cc1xxlfWgyHdnur6QodeF9BsBseOJDL10cgyxQJvYl44CeHAiPuyPe1dpIhY4iXNBdh/AA9O0HI0EsX1oOrPgualYqNOHqm0QKhZI9aWAtMUBBdxLdiyxIPaU8YHvVhmP+P6VALnLW9dwzSL88sqDLJtsgkTR49hoLnJt+k22QV0+0K8YYoGHPeT+1zTIU5oaj4rMquIDy24iFsSHpXEzESDnYwGuJ04H+GFNYoD3ewIu0mQsiD2hfWC5VWPBZTwS+3lw8hAgfPM6DXg1SeM6UvIex72LNB0LXAeJjby+CzwNcD3Mj9xm3n54dpWqsWDW47J/2R0viK2TzZct98wX3ZD712FJua77XfHL5SzAt2xf/ZhssTRZY0nHJW2YtceckeRz7E3LfUiQfjU8nWk8M0/0HjDSs26r7Fdc5rSQ45j0O/p3cZZDBenO/hYt8KSeneQG17eBm4xnl9s3EqXFXZRL6LxovCOnt0bSfLdm/vT5cuZ/4IKAwoXwwirsH/zZa5Gww+XpcaHSpvRF5YSIrz1V++9oVMyJ8tvA5WKEx7UPdPP4laJ644bIEvnlalZ+VvqqyJgA8EBlN4BjQBn5rPESv5osaTwX3VZtAyl/Mm7YFlxcko9PAz0BLtzyJJZY4MTwC6ALOB6grPXxxem/TcTCT2DJasDZwLYWqwYbaRvinuMH5UEgb0ymTlOxUKcPTbQBuXKRcVDaN1FcyeUF6MQSC2JuGR/4bpXxiO+7jEnp/swPGDaZhETGNmOBYyM/TDwDZEmTbVCXD/St7VjgYeh2CclXJFefS1PjUZ5NVX1g2U3EgvhwFm42Bx4FRgBzgO8AfwF+D6wIdAPOAf4PYH6eNBkLYkdoH1huk7HAj5gbAwcD+wEcb2YDjwG3Af2BEwDK7R9fCv/bdCxwHfQ3gIdl9IX18/DtWuAS4GZA1sOuPuCVyrHAMnwkhv2r737XR99Hl7zVqV9n2T5tHlqXe4DTgMeB14AdgbQwPig8A5H8t3DPX5C9CIiYZySSJlczb7ok5lzL7ldc57Q6xrEcdwqznP3lpOUiOxlKHNB8N9/PJ++T0DpEymfZPFjhxP0hH1LCdPPghV96skRsNcvO0l1Q03kafgRA3mzyRSTuCnBxdKxNIUlzCdKc1ytnsQ35a7cRABcYvPcR6QPSJ3ze5eLmROAM4EyfFw3dk3B/ufGcvj0/SeAm9IZ0pvHMgdlXGC/XJC/tgOsLvgUY+jzQ2APg4pqLai6+XWVqorgkrr0A9rmy8jpe5NefbQBudDkpvQzkSWyxcD2MlcO35fIMT+U1EQvrJXX+KFW37ZELBhHGykXykHE9CelNxEKdPlRtg5CxMMngeZZxn3cbWyyU8YH+VRmP+L7LvPAm9LhA5JhH4QEnF8dp4Uc+6vVNMjju5kmTbVCXD/Sv7Vj4EmxYDGCb5M2dyLbKSUhtYjyyVp4kVvWBxTQRC6xnA+Bw3kC4XjHn8RvxvCPAwx/O8ysDOwNXAnnSZCzQjjp8YLlVY8FlPGI9Iu/g5oIEksYr10STeZMI12ouchKUmo6Fh1DnVyzGHYk0OXijznMWnaykqrGQVW5WurQb8znut7F/XSExzrQlSbJefPR9dFlZnfp1lm0lqqFEHr71BNYEuDfIk7WRKTrP4H4NwGx384AtXY6ZNy2dmfPsu1/xmdNCj2M5bjhn+fprLZgN+jrAEz3iGMBXjsYLfJcDva/ciRek7gE5Lz9k6K2XobeuofNUho4kn5Doni0JHtfx0OVmviq296jTpurKne1dl7QRUGLbcNNQl4T0YXUYuVQJQ8v2391R13vAeSXq9HllOpTZDiN9XnLQ5VeiWwG272cc9F1UekFpkIuiRWcG0ugnF+VV5WQUwLJeqVpQ8n4TsWCaynH5A4A+HGtmON7XGQuTYANtywI/jNBugegdiLSqEioWmvChbBuQo1CxsArKknYYVpX85P2mY6GKD1XGI9d5wZzDtsrh+CWjLZbO0XPJCt0GdfvQVixwkcz+z1/K1CGhxqM820L50EQsHAlHZLzZNcOpsYbOtzN0fJJDx0LdPpSNBdfxqIg7KYftNK5I2SO/iVigORw7nwRo/7sAD0t9pUosmHWZ42bM+1ff/e47cJL8LmM6m3HvW7avfky2mBSMx0PVMwC+73IOwL2mrKOzrjLu8io6T+Ceshcg+VfOTbH/hz6J3kF2FWuq737l+qSeKvOy7zjG9a/4drHVC/fEQn+zftFkVrE1HpYyEq4z7l1v5VQ1xKY5q07zi8s2GUrbGulFjdov0RXbjVcLb/tAY/kA6FFYkyr4MPAslN/weSHRlT7g0385YF4KXAZkHVjzF0unADEK+x4XXoMB/vL1EcAmw5G4iS0jI41fumXAz1DJTC7TDlmFyRcwHtp3ovALlIzfZXyoMxa42O2eg5EG4Tsaehcb6W3fNuFD2TYgN6FiYb2E6Fm43prcd9qlig9NjEfmYjZrIb0oSJcDtxdx/1pkjVC3D23EAucAaY8rIuPb1ZyQPjQRC+YvJ17JcNL8pRU3ibFJ3T6UjYUQc8KGIPtHCeHTcOVBY6fJGBg8IDH6RFwnlXCgSiyUqG6+Xwxuk1FAnfvXfkmd0ocyTJibzH0LDye5l5o9NyX/Pz5lsyQf/ZhsSbPQ5DnABFSet+ZmHj/uUe4ARHfQ3JRFFuG5Dn/hThkCcD2SFqZJ33wL9+aaIK2bfvbZr4SY09oexwr9XSzNkOV5ZyONXxM4MfjKY8kLW+Ca9ZParDJtncCmy8O30UnGtrj+1KK0nZF2sXFvu+WhI+WBjy9e/z0D2uZhoNfLhvK9xn2ZW1fuypTd1Dsx+ODbfzcDORwM/wwcAvCXPmnh4dZYgINYbMIYZXwwXnjwdjdgE8bILwDZ/Np0Qqbxz1QHA5wc7qlQMLkfmrz/4wrltPnqLknl9+N6S4OG+MZCg6YtVFWFioVvJqz9Fte3O5TBtnxwjYULweupABe8+wKjgbSsi4SeSeLV6cwInmP2oWws7A5euQbmHHx9BByXMSEWH1xjgRs/kbVwM1EejOsU417KNZJav43VB+GqzD6LpHLTy30LD1Y+Arh2zTogRVaUwrlg78SyW3E9s2UrXfcv5F3mhW1x/1OL3VyPi4Tev/rsd310aW+d+nWWXcZ2aR9eYzkHMG3Kuufajx89uD7hOMD1SPrsY32kLQ9QrgJ8fsjis1+pOqeVHcdcY3UuAQX/8fHXWhQXJtMADsTERUBZuQsvsgyeSPrIZChL/esUvHhOovsBrpumdDkhMZ1ljUnlpR9XRMKHwFSgWzqzg54nw1ZX7sq4NSIpf2aZlx3fmZzUQT+K2t+xyFJqrv2XgxYXLLSXv3wbawE3Vy8D1DEnVDyWkul4i2WNLPX2/C9xAPoVwPIY+2MtuBBp9wGMp9uApoQfAmjXNQUVfhn5nBy+Ayxh0f1vpLGcWyx5ZZNCxgInD7YBx9v0OEb7+gLPA+8DOwBNi2ssZNl1NDLIPzE0S6lkeshYyDOhTh/y6pU8l1jYH8rPAv8AjgHSc9kRSGMbcNO2EhBKQsZCWz64cuEaC+ehQOnzB1gK52aL+W8D/DPaqhKyDcSWpn2QeouuLrFgK+OPSCTn42yZgdLqHo+a8MGVCpdY6InCXgPI+4MAD6TTcjISmD8L4FxYVULHQhs+uHLg0ga2srgOfQ4g7zyMPhAILXXGwpIw9gKA9hM3AdzHtS2TYYDYVLR/aWv/6rvflbnqKAdyfcv21Y/JFgc6WlWZkfTFiRlWDES67EuvtejIXMNx2ezLofcrUk+ZebnKODYcfkms5tUd2t9PUL0sUmQTLgb9DWllJ8OjE8e+/Yma7AmclL+evCP1n4XnHnb1ual85zqA+o8A7EyUQcDjANN5ustDxTzZB5nUPTNPKeK8MtyVcSf0osa0oSkfzDrz7l36L/vZC4D016LrU9ANcdoeclHzIw/76d9BQFPCPjENmA3kxfBzyBfup+B+OMCfsvNXcz8HmMfFWR8glISMhcNhlNjPjwCXAOsBKwN7A/TpHWAvoA1xiYU8u+R9+jg0T7FEXshYyKu+Th/y6pU8l1gYD2XpR7zeAWwNbAhInD+Be/arkBIyFtrywZUP6QdF6xq21x8AtsMc4MtAL6AbcBjADxlvA/sDISRkG4g9Tfsg9RZdXWIhXcbySHgXYHvsl84M+Dw9qWNkwDKlqKZ8kPqKrq6xsD0KYl8n91cA3HCLDMcN44N5oea3OmKhaR9Ah5O4toEUxoN+7qu4ziDn/Ki6PlCH1BELXAcOAziP0X72q1MAjgltCuvvlP2r736XH/MYoxx/isS3bF/9mGwp4qLt/BkwgDEyMceQbZAn4+9/4b5XghNw5bvcd3weMOVwPDCPqLpfKTunVR3HPg3b+Us/8eMl3K8J2CSkv58o/0akvAeIIeaV5E4B+gM+shyU2XAcJIsGxjOg8wZg1iv33Hg/DGQJN9Rc5HIxy3cmJ1c+/x7oDRQJ/ee76xYpRphfhTtfd0bgBfI00/fFAv0mfSgwZV62S/+9AdrST12uHNBCyHQUwvqqLvC3TMpxsZ06rwK2X5YhuTb5Pkpm3Qfn1PDNRCftx2tIvx3YFwhx6Ili5knIWFgapT4HpO3n2Ps0wAPEVYG2xCUW8mzjYYP4tnWeYom8ULFQVHWdPhTVLflFsbAJFKcCwrV5ZTo3By7zIdS8JGQstOWDq8M+scCN4gWALG7fxP0rANuFa5rBQCgJ2QamTU36YNZbdF8UC+n3D0UCeeeaNORHmHQ9dY5HTfmQ9inr2ScWtkMh9wNsA67NOa9JLPAA6AtAKKkrFpr0wZUL1zbgGombcK4puNe7CdgPCL0uQpHzJGQscG/2K4B7D/ah54GfAqsBbUuV/Usb+1ef/e5WIJd8j3ck2adsFumjH5MtjnS0qjYFtbPtGOt5sgcy/wlQl3s8gvdM2w1Iy9JICLVf8Z3Tqo5jqyS2cxykjyaY9izA+cOUkP6a5dZ6z8GRzg2vtZaPC/8ULl8ESBw7DJ9dZBso0cbLXJQXcp0jEq5CH77FSmuT/deHA9lgVz1886mzLd0uVDwbeArgRjBL1kQGf11yFLAL0B+oc2EZOhYWh707AjzkOQTYGOCfVsQiGgvtt0QXTCiKhR7Q2Ro4AGAfHQYwNphel4SOhTZ88OHGNxb6ovCdgSMBjk18Di2h2yBtXxM+pOvMe+5CZlEsmO9zPhgCbGAm1nA/FWVyPVnH3NyUDz60+MbC51D4ngDX6YyJAUA3IKTUHQtN+ODDh0sbnI8C+fGX48/yPoVX0A0ZC7vCjrMBro8GA3Wu7VB849LU/nUbeMbxyXW/ezN0+cGCnBeJb9m++jHZUsRFp+X3gMH8McYhCXjQmbffCrVf6Y96fOblNsYxmLhIKH9ZViPSC7XcDfArV89GavSv5Da8wo09TzdV8hnojewtAP46YWGQWPsvN9Nsh34LQyPAR/45ykfAVyLyV2MhjsbQWGi/HTQWtA3aYCDGeWFhG49iXCPpeNRGNH6yzoUtFj7JQP0pvvtXH/3tYT7X3cc4uuFTNov00Y/JFkc6VE0ZaJeBLlT/MnBRu2ZYaz8RqXMA/ps4KsqAjYEuJMbaf232LqhpZ8Gx14G6/n2SBZW3kH51oTCNhZCMlitLY6EcbyHf6kJhGgshGS1XlsZCOd5CvtWFwjQWQjLqX5a2gT9nnf6G7/7VR39lkDMFGOdIkk/ZLNJHPyZbHOlQNWUgDgZ2hhkfAqPjMGeuFfyzHJ7q1/HnARG5qaYEYCDG/hvArY4qgj+B/ivAP2VYtaMsX7CM1Vhovz01FtpvA1qgsdB+O2gstN8GGgvaBnEwsPBY4bt/9dFfCjROAp4GlnGg1KdsFuejH5MtDlSoijIQHwPDYRJ/ZbZPBKZtCRv4d+yjIrBFTegMBobDzFj6b2cwFt5K/l/S+I8GPwB0D1+8lujIwHDoaSw4klWTmsZCTcR6Fjsc+hoLnqQFVtdYCExoyeKG4z2NhZLkBXpN2yAQkREX47t/9dW/Fr4/AQxw4MC3bF/9mGxxoENVlIE4GVgHZrn8w411Wy//FkHd9Wj5CxYDsfTfBYtVP2/4b0fuDXTze021AzOgsRCY0BLFaSyUIK2GVzQWaiDVs0iNBU/CalLXWKiJWI9itQ08yOpAVd/9q68+/y1N138D3bdsX/2YbOnArqImKwPKgDKgDCgDyoAyoAwoA8qAMqAMKAPKgDKgDCgDyoAyoAwoA8qAMqAMKAPKgDKgDCgDyoAyoAwoA8qAMqAMKAPKgDKgDCgDyoAyoAwoA8qAMqAMKAPKgDKgDCgDyoAyoAwoA8qAMqAMKAPKgDKgDCgDyoAyoAwoA8qAMqAMKAPKgDKgDCgDyoAyoAwoA8qAMqAMKAPKgDKgDCgDyoAyoAwoA8qAMqAMKAPKgDKgDCgDyoAyoAwoA8qAMqAMKAPKgDKgDCgDyoAyoAzExcD/Ax/ZzfZc4XSjAAAAAElFTkSuQmCC" width="623.5" height="19" style="width: 623.5px; height: 19px;"></div><div 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: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; 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: 0px 0px; transform-origin: 0px 0px; ">-----------------------------</div><div 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: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; 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: 0px 0px; transform-origin: 0px 0px; ">We now consider the following congruence:</div><div 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: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; 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: 0px 0px; transform-origin: 0px 0px; "> <img src="data:image/png;base64,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" width="133" height="20" style="width: 133px; height: 20px;"><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: 0px 0px; transform-origin: 0px 0px; "> with 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: 0px 0px; transform-origin: 0px 0px; ">and <img src="data:image/png;base64,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" width="41.5" height="18" style="width: 41.5px; height: 18px;"></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: 0px 0px; transform-origin: 0px 0px; ">The congruence expresses the possibility that the nested sum and product function defined above (Y<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: 0px 0px; transform-origin: 0px 0px; ">), can have a perfect power residue in some modular base. In fact, solving for x<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: 0px 0px; transform-origin: 0px 0px; ">, the congruence always have a trivial solution, namely <img src="data:image/png;base64,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" width="36.5" height="18" style="width: 36.5px; height: 18px;"><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: 0px 0px; transform-origin: 0px 0px; ">, that's because <img src="data:image/png;base64,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" width="58" height="19" style="width: 58px; height: 19px;"><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: 0px 0px; transform-origin: 0px 0px; ">, which is, of course, a perfect power in any modular base. </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: 0px 0px; transform-origin: 0px 0px; ">Given the value of integers N<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: 0px 0px; transform-origin: 0px 0px; ">, 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: 0px 0px; transform-origin: 0px 0px; ">, and certain limit L<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: 0px 0px; transform-origin: 0px 0px; ">, find the sum S<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: 0px 0px; transform-origin: 0px 0px; "> of all positive integer values of <img src="data:image/png;base64,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" width="39" height="18" style="width: 39px; height: 18px;"><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: 0px 0px; transform-origin: 0px 0px; ">, that satisfies the above congruence.</div><div 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: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; 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: 0px 0px; transform-origin: 0px 0px; ">For <img src="data:image/png;base64,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" width="41.5" height="18" style="width: 41.5px; height: 18px;"><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: 0px 0px; transform-origin: 0px 0px; ">, <img src="data:image/png;base64,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" width="44" height="18" style="width: 44px; height: 18px;"><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: 0px 0px; transform-origin: 0px 0px; "> and <img src="data:image/png;base64,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" width="46.5" height="18" style="width: 46.5px; height: 18px;"><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: 0px 0px; transform-origin: 0px 0px; ">, we see that only <img src="data:image/png;base64,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" width="36.5" height="18" style="width: 36.5px; height: 18px;"><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: 0px 0px; transform-origin: 0px 0px; ">and <img src="data:image/png;base64,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" width="36.5" height="18" style="width: 36.5px; height: 18px;"><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: 0px 0px; transform-origin: 0px 0px; "> satisfies the congruence, since: </div><div 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: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; 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: 0px 0px; transform-origin: 0px 0px; "> <img src="data:image/png;base64,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" width="149" height="20" style="width: 149px; height: 20px;"><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: 0px 0px; transform-origin: 0px 0px; ">; and </div><div 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: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; 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: 0px 0px; transform-origin: 0px 0px; "> <img 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" width="582.5" height="20" style="width: 582.5px; height: 20px;"><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: 0px 0px; transform-origin: 0px 0px; ">.</div><div 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: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; 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: 0px 0px; transform-origin: 0px 0px; ">Therefore in this case <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: 0px 0px; transform-origin: 0px 0px; ">S(20,7,3) = 1 + 5 = 6<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: 0px 0px; transform-origin: 0px 0px; ">.</div><div 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: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; 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: 0px 0px; transform-origin: 0px 0px; ">For <img src="data:image/png;base64,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" width="41.5" height="18" style="width: 41.5px; height: 18px;"><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: 0px 0px; transform-origin: 0px 0px; ">, <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGcAAAAkCAYAAACUjSQ7AAAAAXNSR0IArs4c6QAAAERlWElmTU0AKgAAAAgAAYdpAAQAAAABAAAAGgAAAAAAA6ABAAMAAAABAAEAAKACAAQAAAABAAAAZ6ADAAQAAAABAAAAJAAAAAAk0t2MAAAE5UlEQVRoBe2YW6hVVRSGrWOmeekiRcdrhlpJWD2oEMoRFLOCQBTJlDhgaC/Rk1gUEnQRH3oSHwyzgqAHrURSM8yOFmqRUYr0UkkadtHyekxLre9friHDyd57zX2WyNnnzB/+Pccel7nWHGPNy1o9eiSkDKQMpAykDKQMpAx0+wxc0+0zUH8CriXkCdgbrooMb8ZvFhwJz8MtsA2ehqWwguhjFTgmstcm/NoqxH8SGd9Z3FSUx+H38D/4LozBMzidhRdgG9wBFX8GPglLYxg9fA7VqfHhyF6fdzGK1Y3OhD1hI0Cri+53L7Sxq40pznz8VBRxCjQ8iqA+NIvmmbJM+zbB/uYWRHQ2Fh8Vw8fFLgUR3V8Vl0e4yjqoJK6HNpai4kzFV8mX/0oY4k0Usp2DD4TGev8fIsBuTO3LBR30wv5tEKM4PTWNCu0bloOi4nzkfEdVGPBoZ1ehOoz7iLSbsvadgt5eyWP+cLGaRX0L4jqzebobS63iNOOnGaFc6aGuht8wyEcHg1tCJ21yMbD9ZbVzHurkUByP4jm4B37ljF8gt7v/XVWcy8Ca8sFtrzFIs/XBZ07oV09xThC8xnVQrTi6kGaVNsGn4ERo2GxCF2+11xp0uquGfc5wt5MzMaY4A/B8EG6F+7Ooiz9DnOzF1/ijC6m9Dt4IDd2lOINtwLR6DamG484wzMmZ2DNUVPg/FZ38lNiDzt4b+VZ42OlakJ+FOgi8Cl+ABq2v39mfOtsJ+G+sM6aS+26U0yoZrrBukOuvVnG8rUPFsf1GL43auP6CtnlpabPi9EN+C2ojbIX/woegocxLpx4Ou6b115FWq8DVwO3uIn52OHUmnnSK25yciTEzZzqeP8Cf8mDNHkuUivNNrn+ddgRcAjVDboLjoKHMkqa1eaF1VKI9UCK2ntC/cdb4Bb3EVoPPvx0gLvl64yWlE+5F1t6ywulUHB2tBTsUaHYtgCrUUihMgXZBHRfLzBxN/zdgo+B3brQ5v1m/54b3r8OTQTGXoehAoKQLPrH+6dM6eTNcBf+BrVDLmuDXdhXtSKbtHj8+0TaDKo3c234NHYpmjoqjveMzF6iZY9DMWQ4HwRehvj8ZfHHKLGnWXyO1v7ib9QVw6kz0tkOhsVZx+uM8Ee6EfuPyxdFXWuFruCyTLv6MprnD/f/YyR0RG+20tolBzs8HaltApXGPdUo/ATJ1reLoydd7SvjU++Kok7OwFdpyhnjZkqaXVxW4DHSfdggp08+AMsF1xOrY3w71qWoS1KFA+66HdC25QqfgD71Rcq3izMydtwRBYXFewq7TlIdOeIZtCL5wpq+n7SynNSU0BjqtbYCz4UCoGRK+492Pzh64dcinYBR05pazqn19ENGL/xdy25e0TYFdsTocKFZcDLsKWhmIjWttwaD0JfrP3F9fqENo6VNfx6BOxVG4Aa8PoN3E08gqiIdOFno67vFK5OHwU2ixat+DWh4bHXcyAD39Nja9fN9VMKgW7Fr2FbMI6kEX9VFYujNwMoyCLqak2w1Yq1nUx/WgL826mMf7/DH/sG3Htto7N5Cs72Raym218GOTbj9cCKthBoafoeKO5pQs3WOwKmLX0KodJENUBrRyjIcjc+8faXfBsntx3l1qUgZSBlIGUgZSBlIGUgZSBlIGUgZSBlIGUgZKZuB/nRIZFJN8DtIAAAAASUVORK5CYII=" width="51.5" height="18" style="width: 51.5px; height: 18px;"><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: 0px 0px; transform-origin: 0px 0px; "> and <img src="data:image/png;base64,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" width="46.5" height="18" style="width: 46.5px; height: 18px;"><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: 0px 0px; transform-origin: 0px 0px; ">, the above equation is satisfied, <img src="data:image/png;base64,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" width="121.5" height="19" style="width: 121.5px; height: 19px;"><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: 0px 0px; transform-origin: 0px 0px; ">.</div><div 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: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; 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: 0px 0px; transform-origin: 0px 0px; ">Therefore: <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: 0px 0px; transform-origin: 0px 0px; ">S(20,10,3) = sum(1:20) = 210<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: 0px 0px; transform-origin: 0px 0px; ">.</div></div></div>

For a positive integer , we define the function , as follows:
                        ; and
                         for .
Hence, for  we have:
                        
And if  :
                        
-----------------------------
We now consider the following congruence:
                         with and 
The congruence expresses the possibility that the nested sum and product function defined above (), can have a perfect power residue in some modular base. In fact, solving for , the congruence always have a trivial solution, namely , that's because , which is, of course, a perfect power in any modular base. 
Given the value of integers , , and certain limit , find the sum  of all positive integer values of , that satisfies the above congruence.
For ,  and , we see that only and  satisfies the congruence, since: 
                        ;  and             
                        .
Therefore in this case S(20,7,3) = 1 + 5 = 6.
For ,  and , the above equation is satisfied, .
Therefore: S(20,10,3) = sum(1:20) = 210.

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David Hill on 1 Jan 2023

I am getting all test solutions correct except: test2, test5, and test8.

For test2 for example, I am only getting [1 5] = 6.

Ramon Villamangca on 2 Jan 2023

Hi David, test suites has been corrected. Please try again. Thanks.

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Problem 55595. Easy Sequences 87: Perfect Power Modular Residue of a Nested Sum-product Function

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Problem 55595. Easy Sequences 87: Perfect Power Modular Residue of a Nested Sum-product Function

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