View source for 27.12

{{DISPLAYTITLE:Functions of Number Theory - 27.12 Asymptotic Formulas: Primes}}

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| [https://dlmf.nist.gov/27.12.E1 27.12.E1] || <math qid="Q8085">\lim_{n\to\infty}\frac{p_{n}}{n\ln@@{n}} = 1</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\lim_{n\to\infty}\frac{p_{n}}{n\ln@@{n}} = 1</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>limit((p[n])/(n*ln(n)), n = infinity) = 1</syntaxhighlight> || <syntaxhighlight lang=mathematica>Limit[Divide[Subscript[p, n],n*Log[n]], n -> Infinity, GenerateConditions->None] == 1</syntaxhighlight> || Failure || Failure || Skip - No test values generated || <div class="toccolours mw-collapsible mw-collapsed">Failed [10 / 10]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -1.0
Test Values: {Rule[Subscript[p, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -1.0
Test Values: {Rule[Subscript[p, n], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/27.12.E2 27.12.E2] || <math qid="Q8086">p_{n} > n\ln@@{n}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>p_{n} > n\ln@@{n}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>p[n] > n*ln(n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[p, n] > n*Log[n]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [6 / 10]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 3.295836867 < -1.500000000
Test Values: {p[n] = -3/2, n = 3}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 3.295836867 < 1.500000000
Test Values: {p[n] = 3/2, n = 3}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [25 / 30]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Greater[Complex[0.8660254037844387, 0.49999999999999994], 0.0]
Test Values: {Rule[n, 1], Rule[Subscript[p, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Greater[Complex[0.8660254037844387, 0.49999999999999994], 1.3862943611198906]
Test Values: {Rule[n, 2], Rule[Subscript[p, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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