Problem 46117. Test approximations of the prime counting function

Created by ChrisR

Solve Later

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That is, the problem seeks the inverse of the prime counting function </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=\"pi(n)\"/></w:customXmlPr><w:r><w:t>\\pi(n)</w:t></w:r></w:customXml><w:r><w:t>, which provides the number of primes less than or equal to </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>. The </w:t></w:r><w:hyperlink w:docLocation=\"https://mathworld.wolfram.com/PrimeNumberTheorem.html\"><w:r><w:t>Prime Number Theorem</w:t></w:r></w:hyperlink><w:r><w:t> gives approximate forms 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=\"pi(n)\"/></w:customXmlPr><w:r><w:t>\\pi(n)</w:t></w:r></w:customXml><w:r><w:t> for large </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>. Two such approximations are </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/ln(n)\"/></w:customXmlPr><w:r><w:t>n/\\ln(n)</w:t></w:r></w:customXml><w:r><w:t> and the </w:t></w:r><w:hyperlink w:docLocation=\"https://en.wikipedia.org/wiki/Logarithmic_integral_function#Offset_logarithmic_integral\"><w:r><w:t>offset logarithmic integral</w:t></w:r></w:hyperlink><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=\"Li(n) = li(n) - li(2)\"/></w:customXmlPr><w:r><w:t>{\\rm Li(n)} ={\\rm li(n)} - {\\rm li(2)}</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=\"li(x) = integral of 1/ln(t) from t = 0 to t = infinity\"/></w:customXmlPr><w:r><w:t>{\\rm li}(x) = \\int_0^\\infty dt/\\ln(t)</w:t></w:r></w:customXml><w:r><w:t> (See </w:t></w:r><w:hyperlink w:docLocation=\"https://www.mathworks.com/matlabcentral/cody/problems/46066-evaluate-the-logarithmic-integral\"><w:r><w:t>Cody Problem 46066</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>Test these approximations by computing two ratios: </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=\"r1 = pi(n)/[n/ln(n)]\"/></w:customXmlPr><w:r><w:t>r_1 = \\pi(n)/[n/\\ln(n)]</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=\"r2 = pi(n)/Li(n)\"/></w:customXmlPr><w:r><w:t>r_2 = \\pi(n)/{\\rm Li}(n)</w:t></w:r></w:customXml><w:r><w:t>. Do not round the approximations to integers. For </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 = 100\"/></w:customXmlPr><w:r><w:t>n = 100</w:t></w:r></w:customXml><w:r><w:t>, you will find that the first approximation is about 13% low and the second is about 16% high. However, for </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 = 10^8\"/></w:customXmlPr><w:r><w:t>n = 10^8</w:t></w:r></w:customXml><w:r><w:t>, the first approximation is 6% low and the second is only 0.01% high. </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: 162.817px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 81.4083px; transform-origin: 407px 81.4083px; vertical-align: baseline; "><div style="block-size: 90px; 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 45px; text-align: left; transform-origin: 384px 45px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><a target='_blank' href = "https://www.mathworks.com/matlabcentral/cody/problems/241">Cody Problem 241</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: 61.4583px 7.79167px; transform-origin: 61.4583px 7.79167px; unicode-bidi: normal; ">, which is based on <a target='_blank' href = "http://projecteuler.net/problem=7">Project Euler Problem 7</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: 73.1083px 7.79167px; transform-origin: 73.1083px 7.79167px; unicode-bidi: normal; ">, asks us to identify 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: 91px 7.79167px; transform-origin: 91px 7.79167px; unicode-bidi: normal; ">th prime number. That is, the problem seeks the inverse of the prime counting function <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAD4AAAAlCAYAAAAa0FocAAACX0lEQVRoge2YXZGDMBSFPw84wAAGUFAFdYADHNRCNSChHmoBDVjoPoSzvaVJubCkywNnhtmZpHv/cnJuEjhw4MCBPKiAYiNb5Ya2suIMXDe0VwG38e9usXXSQgn07DT5EyG4XLSsMttfhRIYgDqznwvQZfaxCFfCPsyNglDg0xd8zaIEHuRfbeFKoPy/49uBVHy30EkM5FHyrD5LQgtqHV+sldSE6p8X+KzH31t7JdB88DPFjZUsKwgVeyz4yoidFh/trsA9Ya+djA8JXzG/i1pbMQbREyonh7fxU4C9GUvRSsXzBFBMfOn/b4REL2Z+TrW9BX/BjUAreAqFbUXNONY6bT2cfqX+D560tj1ZyQzMF/KEr0C/KHjdQ1qxxoz145h3r3kTP/NM/Mz7KUy2POcBaYtncd5gqac9JQZ4hWNJ4iryMNqf0nTgfRFS+FPiorStsPbZxWljSeJKLBZwbeY8TFvTTX4hStsgFJxXNDp8iYtJKTa1H+ZiWCVuEK+wBMMjLtMA5lbJtqzYKqmTeA8lsjfX9t4git7NWBcZu8wY96qrEhsic1ZrZGcuoS5hK4mSZ4LT6ov6olvDaxFiUNCfRMYmFltRqb2YVjn89iy4nlp6T1eWyZw+zx7qZgIVK1L2LNP02PBp60gv3MJm++gQMW6Z4G0r1m6KnraNxbTDHp09jxktC2kOofpNIsiC54lqqWj0pFugLkGpFSrG+QafqPas7N85UOO7XPwVDTt5hLDI/R6md71dvrR2+LVhKe6sPKl9A7rnbx1gDptZ0LLd+/eJHbyvHThw4MC/4wdzuO6gMG8oWgAAAABJRU5ErkJggg==" alt="pi(n)" style="width: 31px; height: 18.5px;" width="31" 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: 185.15px 7.79167px; transform-origin: 185.15px 7.79167px; unicode-bidi: normal; ">, which provides the number of primes less than or equal to 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: 17.8833px 7.79167px; transform-origin: 17.8833px 7.79167px; unicode-bidi: normal; ">. The <a target='_blank' href = "https://mathworld.wolfram.com/PrimeNumberTheorem.html">Prime Number Theorem</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: 87.9px 7.79167px; transform-origin: 87.9px 7.79167px; unicode-bidi: normal; "> gives approximate forms of <img src="data:image/png;base64,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" alt="pi(n)" style="width: 31px; height: 18.5px;" width="31" 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: 29.5583px 7.79167px; transform-origin: 29.5583px 7.79167px; unicode-bidi: normal; "> for large 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: 96.8583px 7.79167px; transform-origin: 96.8583px 7.79167px; unicode-bidi: normal; ">. Two such approximations are <img src="data:image/png;base64,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" alt="n/ln(n)" style="width: 52px; height: 18.5px;" width="52" 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: 27.225px 7.79167px; transform-origin: 27.225px 7.79167px; unicode-bidi: normal; "> and the <a target='_blank' href = "https://en.wikipedia.org/wiki/Logarithmic_integral_function#Offset_logarithmic_integral">offset logarithmic integral</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: 1.94167px 7.79167px; transform-origin: 1.94167px 7.79167px; unicode-bidi: normal; "> <img src="data:image/png;base64,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" alt="Li(n) = li(n) - li(2)" style="width: 126px; height: 18.5px;" width="126" 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: 24.8917px 7.79167px; transform-origin: 24.8917px 7.79167px; unicode-bidi: normal; ">, where <img src="data:image/png;base64,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" alt="li(x) = integral of 1/ln(t) from t = 0 to t = infinity" style="width: 122px; height: 27px;" width="122" height="27"><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 7.79167px; transform-origin: 18.6667px 7.79167px; unicode-bidi: normal; "> (See <a target='_blank' href = "https://www.mathworks.com/matlabcentral/cody/problems/46066-evaluate-the-logarithmic-integral">Cody Problem 46066</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: 4.275px 7.79167px; transform-origin: 4.275px 7.79167px; unicode-bidi: normal; ">).</div><div style="block-size: 63.8167px; 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 31.9083px; text-align: left; transform-origin: 384px 31.9083px; 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: 160.642px 7.79167px; transform-origin: 160.642px 7.79167px; unicode-bidi: normal; ">Test these approximations by computing two ratios: <img src="data:image/png;base64,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" alt="r1 = pi(n)/[n/ln(n)]" style="width: 128px; height: 20px;" width="128" height="20"><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="r2 = pi(n)/Li(n)" style="width: 100.5px; height: 20px;" width="100.5" height="20"><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: 57.95px 7.79167px; transform-origin: 57.95px 7.79167px; unicode-bidi: normal; ">. Do not round the approximations to integers. For <img src="data:image/png;base64,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" alt="n = 100" style="width: 51.5px; height: 18px;" width="51.5" 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: 257.5px 7.79167px; transform-origin: 257.5px 7.79167px; unicode-bidi: normal; ">, you will find that the first approximation is about 13% low and the second is about 16% high. However, for <img src="data:image/png;base64,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" alt="n = 10^8" style="width: 49px; height: 19px;" width="49" 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: 216.658px 7.79167px; transform-origin: 216.658px 7.79167px; unicode-bidi: normal; ">, the first approximation is 6% low and the second is only 0.01% high. </div></div></div>

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ChrisR on 3 Jan 2021 at 15:27

Test added to discourage lookup table solutions.

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Problem 46117. Test approximations of the prime counting function

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Problem 46117. Test approximations of the prime counting function

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