Problem 52183. List the nth prime quartet prefix

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>Prime numbers larger than 5 can end only in 1, 3, 7, or 9, but of course not all numbers ending in these four digits are prime. Let’s call a group of four prime numbers 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=\"10k\"/></w:customXmlPr><w:r><w:t>10k</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=\"10(k+1)\"/></w:customXmlPr><w:r><w:t>10(k+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\"/></w:customXmlPr><w:r><w:t>k</w:t></w:r></w:customXml><w:r><w:t> an integer 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=\"k &gt;= 1\"/></w:customXmlPr><w:r><w:t>k\\ge 1</w:t></w:r></w:customXml><w:r><w:t>) a </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>prime quartet</w:t></w:r><w:r><w:t>. The first one is 11, 13, 17, and 19, and the second is 101, 103, 107, and 109. Therefore, the prefix (or the number excluding the last digit) of the first prime quartet is 1, and the prefix of the second prime quartet is 10.</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 returns 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 prime quartet prefix. </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: Prove that the sequence of prime quartet prefixes is infinite. Just drop your proof in the comments below. </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: 144px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407.5px 72px; transform-origin: 407.5px 72px; 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: 384.5px 42px; text-align: left; transform-origin: 384.5px 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: 384.5px 7.66667px; transform-origin: 384.5px 7.66667px; unicode-bidi: normal; "><span style="">Prime numbers larger than 5 can end only in 1, 3, 7, or 9, but of course not all numbers ending in these four digits are prime. Let’s call a group of four prime numbers between </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" alt="10k" style="width: 25px; height: 18px;" width="25" height="18"></span><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 7.66667px; transform-origin: 15.5583px 7.66667px; unicode-bidi: normal; "><span style=""> and </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" alt="10(k+1)" style="width: 61px; height: 18.5px;" width="61" height="18.5"></span><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: 18.6667px 7.66667px; transform-origin: 18.6667px 7.66667px; unicode-bidi: normal; "><span style=""> (with </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">k</span><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: 48.625px 7.66667px; transform-origin: 48.625px 7.66667px; unicode-bidi: normal; "><span style=""> an integer and </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" alt="k >= 1" style="width: 36px; height: 18px;" width="36" height="18"></span><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: 10.1083px 7.66667px; transform-origin: 10.1083px 7.66667px; unicode-bidi: normal; "><span style="">) a </span></span><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: 41.2333px 7.66667px; transform-origin: 41.2333px 7.66667px; unicode-bidi: normal; "><span style="font-style: italic; ">prime quartet</span></span><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: 30.7583px 7.66667px; transform-origin: 30.7583px 7.66667px; unicode-bidi: normal; "><span style="">. The first one is 11, 13, 17, and 19, and the second is 101, 103, 107, and 109. Therefore, the prefix (or the number excluding the last digit) of the first prime quartet is 1, and the prefix of the second prime quartet is 10.</span></span></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: 384.5px 10.5px; text-align: left; transform-origin: 384.5px 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: 99.4333px 7.66667px; transform-origin: 99.4333px 7.66667px; unicode-bidi: normal; "><span style="">Write a function that returns the </span></span><span style="font-family: &quot;STIXGeneral&quot;, &quot;STIXGeneral-webfont&quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">n</span><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: 71.95px 7.66667px; transform-origin: 71.95px 7.66667px; unicode-bidi: normal; "><span style="">th prime quartet prefix. </span></span></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: 384.5px 10.5px; text-align: left; transform-origin: 384.5px 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: 355.5px 7.66667px; transform-origin: 355.5px 7.66667px; unicode-bidi: normal; "><span style="">Optional: Prove that the sequence of prime quartet prefixes is infinite. Just drop your proof in the comments below. </span></span></div></div></div>

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Last Solution submitted on May 02, 2023

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Nikolaos Nikolaou on 11 Apr 2023

Nice problem , made me think :)

Augusto Mazzei on 2 May 2023

Ah! this was a cool opportunity to use a regex in a fair way:

with 'n' input
persistent a
if isempty(a)
primes(27e7);
a=regexp(sprintf('%d.',ans),'(\d+)1\.\13\.\17\.\19\.','match');
end
strsplit(a{n},'.'); str2num(ans{1}(1:end-1))

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Problem 52183. List the nth prime quartet prefix

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