Problem 45964. Compute the nth Pythagorean prime

Created by ChrisR

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{"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>Pythagorean primes have the form </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 = 4n+1\"/></w:customXmlPr><w:r><w:t>p = 4n+1</w:t></w:r></w:customXml><w:r><w:t>, where </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=\"n\"/></w:customXmlPr><w:r><w:t>n</w:t></w:r></w:customXml><w:r><w:t> is an integer, and they can be written as the sum of squares of two integers. More information is available at</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://en.wikipedia.org/wiki/Pythagorean_prime\"><w:r><w:t>Wikipedia</w:t></w:r></w:hyperlink><w:r><w:t>,</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://www.youtube.com/watch?v=yu_aqA7mw7E\"><w:r><w:t>Numberphile</w:t></w:r></w:hyperlink><w:r><w:t>, and the</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://oeis.org/A002144\"><w:r><w:t>Online Encyclopedia of Integer Sequences</w:t></w:r></w:hyperlink><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>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=\"n\"/></w:customXmlPr><w:r><w:t>n</w:t></w:r></w:customXml><w:r><w:t>th Pythagorean prime </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> and two integers </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=\"a\"/></w:customXmlPr><w:r><w:t>a</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:attr w:name=\"altTextString\" w:val=\"b\"/></w:customXmlPr><w:r><w:t>b</w:t></w:r></w:customXml><w:r><w:t> such that </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 = a^2+b^2\"/></w:customXmlPr><w:r><w:t>p = a^2+b^2</w:t></w:r></w:customXml><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: 20.4333px; 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: 72px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 36px; transform-origin: 407px 36px; 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; "><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: 109.3px 7.8px; transform-origin: 109.3px 7.8px; unicode-bidi: normal; ">Pythagorean primes have the form <img src="data:image/png;base64,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" alt="p = 4n+1" style="width: 70px; height: 18px;" width="70" 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: 24.9px 7.8px; transform-origin: 24.9px 7.8px; unicode-bidi: normal; ">, where 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: 209.25px 7.8px; transform-origin: 209.25px 7.8px; unicode-bidi: normal; "> is an integer, and they can be written as the sum of squares of two integers. More information is available at<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: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> <a target='_blank' href = "https://en.wikipedia.org/wiki/Pythagorean_prime">Wikipedia</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: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; ">,<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: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> <a target='_blank' href = "https://www.youtube.com/watch?v=yu_aqA7mw7E">Numberphile</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: 27.2333px 7.8px; transform-origin: 27.2333px 7.8px; unicode-bidi: normal; ">, and 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: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> <a target='_blank' href = "https://oeis.org/A002144">Online Encyclopedia of Integer Sequences</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: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; ">.</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: 42.0167px 7.8px; transform-origin: 42.0167px 7.8px; unicode-bidi: normal; ">Compute the 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: 68.85px 7.8px; transform-origin: 68.85px 7.8px; unicode-bidi: normal; ">th Pythagorean prime 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: 55.2333px 7.8px; transform-origin: 55.2333px 7.8px; unicode-bidi: normal; "> and two integers 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: 15.5667px 7.8px; transform-origin: 15.5667px 7.8px; unicode-bidi: normal; "> and b<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: 32.2833px 7.8px; transform-origin: 32.2833px 7.8px; unicode-bidi: normal; "> such that <img src="data:image/png;base64,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" alt="p = a^2+b^2" style="width: 72.5px; height: 19px;" width="72.5" height="19"><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: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; ">.</div></div></div>

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Problem 45964. Compute the nth Pythagorean prime

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