Problem 47950. Test the generalized Legendre conjecture

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:r><w:t>The Legendre conjecture states that for every integer </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> there is a prime number between </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^2\"/></w:customXmlPr><w:r><w:t>n^2</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=\"(n+1)^2\"/></w:customXmlPr><w:r><w:t>(n+1)^2</w:t></w:r></w:customXml><w:r><w:t>. The generalized Legendre conjecture (GLC) is that there is a prime number between </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^K\"/></w:customXmlPr><w:r><w:t>n^K</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=\"(n+1)^K\"/></w:customXmlPr><w:r><w:t>(n+1)^K</w:t></w:r></w:customXml><w:r><w:t>; a further conjecture is that the smallest </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=\"K\"/></w:customXmlPr><w:r><w:t>K</w:t></w:r></w:customXml><w:r><w:t> possible is </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=\"log(1151)/log(95)\"/></w:customXmlPr><w:r><w:t>\\log(1151)/\\log(95)</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>Write a function that takes a value of </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=\"K\"/></w:customXmlPr><w:r><w:t>K</w:t></w:r></w:customXml><w:r><w:t>, which you can assume to be less than </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=\"log(1151)/log(95)\"/></w:customXmlPr><w:r><w:t>\\log(1151)/\\log(95)</w:t></w:r></w:customXml><w:r><w:t>, and determines the first value of </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> for which the GLC fails as well as the interval [</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^K\"/></w:customXmlPr><w:r><w:t>n^K</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:attr w:name=\"altTextString\" w:val=\"(n+1)^K\"/></w:customXmlPr><w:r><w:t>(n+1)^K</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: 114px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 57px; transform-origin: 407px 57px; vertical-align: baseline; "><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: 166.475px 7.79167px; transform-origin: 166.475px 7.79167px; unicode-bidi: normal; ">The Legendre conjecture states that for every integer 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: 106.192px 7.79167px; transform-origin: 106.192px 7.79167px; unicode-bidi: normal; "> there is a prime number between <img src="data:image/png;base64,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" alt="n^2" style="width: 15.5px; height: 19px;" width="15.5" 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: 15.5583px 7.79167px; transform-origin: 15.5583px 7.79167px; unicode-bidi: normal; "> and <img src="data:image/png;base64,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" alt="(n+1)^2" style="width: 51.5px; height: 19.5px;" width="51.5" height="19.5"><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: 56.0167px 7.79167px; transform-origin: 56.0167px 7.79167px; unicode-bidi: normal; ">. The generalized Legendre conjecture (GLC) is that there is a prime number between <img src="data:image/png;base64,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" alt="n^K" style="width: 18px; height: 19px;" width="18" 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: 15.5583px 7.79167px; transform-origin: 15.5583px 7.79167px; unicode-bidi: normal; "> and <img src="data:image/png;base64,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" alt="(n+1)^K" style="width: 54.5px; height: 19.5px;" width="54.5" height="19.5"><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: 98.3917px 7.79167px; transform-origin: 98.3917px 7.79167px; unicode-bidi: normal; ">; a further conjecture is that the smallest K<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: 36.5667px 7.79167px; transform-origin: 36.5667px 7.79167px; unicode-bidi: normal; "> possible is <img src="data:image/png;base64,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" alt="log(1151)/log(95)" style="width: 121.5px; height: 18.5px;" width="121.5" 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: 3.88333px 7.79167px; transform-origin: 3.88333px 7.79167px; unicode-bidi: normal; ">. </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: 114.992px 7.79167px; transform-origin: 114.992px 7.79167px; unicode-bidi: normal; ">Write a function that takes a value of K<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: 124.075px 7.79167px; transform-origin: 124.075px 7.79167px; unicode-bidi: normal; ">, which you can assume to be less than <img src="data:image/png;base64,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" alt="log(1151)/log(95)" style="width: 121.5px; height: 18.5px;" width="121.5" 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: 77.0167px 7.79167px; transform-origin: 77.0167px 7.79167px; unicode-bidi: normal; ">, and determines the first value of 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: 145.075px 7.79167px; transform-origin: 145.075px 7.79167px; unicode-bidi: normal; "> for which the GLC fails as well as the interval [<img src="data:image/png;base64,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" alt="n^K" style="width: 18px; height: 19px;" width="18" 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: 1.94167px 7.79167px; transform-origin: 1.94167px 7.79167px; unicode-bidi: normal; "> <img src="data:image/png;base64,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" alt="(n+1)^K" style="width: 54.5px; height: 19.5px;" width="54.5" height="19.5"><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: 3.88333px 7.79167px; transform-origin: 3.88333px 7.79167px; unicode-bidi: normal; ">].</div></div></div>

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Problem 47950. Test the generalized Legendre conjecture

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Problem 47950. Test the generalized Legendre conjecture

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