Problem 51758. List Proth primes

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>As of datenum 738289, three of the </w:t></w:r><w:hyperlink w:docLocation=\"https://en.wikipedia.org/wiki/Largest_known_prime_number#The_twenty_largest_known_prime_numbers\"><w:r><w:rPr><w:u/></w:rPr><w:t>twenty largest known prime numbers</w:t></w:r></w:hyperlink><w:r><w:t> are Proth primes, prime numbers of 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=\"k 2^m+1\"/></w:customXmlPr><w:r><w:t>k\\cdot 2^m + 1</w:t></w:r></w:customXml><w:r><w:t> with </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 &lt;= 2^m\"/></w:customXmlPr><w:r><w:t>k \\le 2^m</w:t></w:r></w:customXml><w:r><w:t>. For example, taking </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 = 1\"/></w:customXmlPr><w:r><w:t>k = 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: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> gives 3, the first Proth prime, and taking </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 = 3\"/></w:customXmlPr><w:r><w:t>k = 3</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=\"m = 5\"/></w:customXmlPr><w:r><w:t>m = 5</w:t></w:r></w:customXml><w:r><w:t> gives 97, the sixth Proth prime. The number 199 is prime but not a Proth prime because </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 = 99 &gt; 2^m = 2\"/></w:customXmlPr><w:r><w:t>k = 99 &gt; 2^m = 2</w:t></w:r></w:customXml><w:r><w:t>. The number 49 is a Proth number (</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 = 3\"/></w:customXmlPr><w:r><w:t>k = 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:attr w:name=\"altTextString\" w:val=\"m = 4\"/></w:customXmlPr><w:r><w:t>m = 4</w:t></w:r></w:customXml><w:r><w:t>) but not prime.</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 list the Proth primes between two limits </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>. Also provide the values 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> 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=\"m\"/></w:customXmlPr><w:r><w:t>m</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>Optional: Values 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> for which no values 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 2^m + 1\"/></w:customXmlPr><w:r><w:t>k\\cdot 2^m + 1</w:t></w:r></w:customXml><w:r><w:t> are prime are called Sierpinski numbers. Show that 78,557 is the smallest Sierpinski number. For more, see </w:t></w:r><w:hyperlink w:docLocation=\"https://en.wikipedia.org/wiki/Sierpi%C5%84ski_number#Sierpi%C5%84ski_problem\"><w:r><w:rPr><w:u/></w:rPr><w:t>this page</w:t></w:r></w:hyperlink><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: 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: 84px; 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 42px; text-align: left; transform-origin: 384px 42px; 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: 111.625px 7.91667px; transform-origin: 111.625px 7.91667px; unicode-bidi: normal; ">As of datenum 738289, three of the <a target='_blank' href = "https://en.wikipedia.org/wiki/Largest_known_prime_number#The_twenty_largest_known_prime_numbers">twenty largest known prime numbers</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: 141.567px 7.91667px; transform-origin: 141.567px 7.91667px; unicode-bidi: normal; "> are Proth primes, prime numbers of the form <img src="data:image/png;base64,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" alt="k 2^m+1" style="width: 60px; height: 19.5px;" width="60" 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: 16.3333px 7.91667px; transform-origin: 16.3333px 7.91667px; unicode-bidi: normal; "> with <img src="data:image/png;base64,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" alt="k <= 2^m" style="width: 43.5px; height: 19.5px;" width="43.5" 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: 67.2833px 7.91667px; transform-origin: 67.2833px 7.91667px; unicode-bidi: normal; ">. For example, taking <img src="data:image/png;base64,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" alt="k = 1" style="width: 36px; height: 18px;" width="36" 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: 15.5583px 7.91667px; transform-origin: 15.5583px 7.91667px; unicode-bidi: normal; "> and <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: 127.175px 7.91667px; transform-origin: 127.175px 7.91667px; unicode-bidi: normal; "> gives 3, the first Proth prime, and taking <img src="data:image/png;base64,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" alt="k = 3" style="width: 36px; height: 18px;" width="36" 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: 15.5583px 7.91667px; transform-origin: 15.5583px 7.91667px; unicode-bidi: normal; "> and <img src="data:image/png;base64,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" alt="m = 5" 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: 1.94167px 7.91667px; transform-origin: 1.94167px 7.91667px; unicode-bidi: normal; "> gives 97, the sixth Proth prime. The number 199 is prime but not a Proth prime because <img src="data:image/png;base64,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" alt="k = 99 > 2^m = 2" style="width: 103px; height: 19.5px;" width="103" 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: 59.075px 7.91667px; transform-origin: 59.075px 7.91667px; unicode-bidi: normal; ">. The number 49 is a Proth number (<img src="data:image/png;base64,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" alt="k = 3" style="width: 36px; height: 18px;" width="36" 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: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; ">, <img src="data:image/png;base64,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" alt="m = 4" 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: 47.0583px 7.91667px; transform-origin: 47.0583px 7.91667px; unicode-bidi: normal; ">) but not prime.</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: 181.117px 7.91667px; transform-origin: 181.117px 7.91667px; unicode-bidi: normal; ">Write a function to list the Proth primes between two limits 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: 15.5583px 7.91667px; transform-origin: 15.5583px 7.91667px; 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: 85.9583px 7.91667px; transform-origin: 85.9583px 7.91667px; unicode-bidi: normal; ">. Also provide the values 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: 15.5583px 7.91667px; transform-origin: 15.5583px 7.91667px; unicode-bidi: normal; "> and 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: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; 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: 60.55px 7.91667px; transform-origin: 60.55px 7.91667px; unicode-bidi: normal; ">Optional: Values 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: 71.5667px 7.91667px; transform-origin: 71.5667px 7.91667px; unicode-bidi: normal; "> for which no values of <img src="data:image/png;base64,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" alt="k 2^m + 1" style="width: 60px; height: 19.5px;" width="60" 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: 204.217px 7.91667px; transform-origin: 204.217px 7.91667px; unicode-bidi: normal; "> are prime are called Sierpinski numbers. Show that 78,557 is the smallest Sierpinski number. For more, see <a target='_blank' href = "https://en.wikipedia.org/wiki/Sierpi%C5%84ski_number#Sierpi%C5%84ski_problem">this page</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: 3.88333px 7.91667px; transform-origin: 3.88333px 7.91667px; unicode-bidi: normal; ">. </div></div></div>

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Last Solution submitted on Jun 17, 2022

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Dyuman Joshi on 2 Jun 2022

Chris, what if there are multiple pairs of (k,m) that result in a particular proth number, which pair should we report?

For example -

k=4, m=2, p=17
k=2, m=3, p=17
k=1, m=4, p=17

also
k=1, m=2, p=5
k=2, m=1, p=5

ChrisR on 7 Jun 2022 at 13:31

Good question, Dyuman. Please use the largest m possible.

Dyuman Joshi on 17 Jun 2022 at 9:56

Thanks for the clarification, Chris.
Nice question!

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Problem 51758. List Proth primes

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