where <math>\pi(x)</math> is the number of primes up to and including <math>x</math>. This means that for any number <math>x</math>, the number of primes up to and including <math>x</math> approaches <math>x</math> divided by the the log to base e (or the natural log) of <math>x</math> as <math>x</math> approaches infinity and <math>{x/\ln(x)}</math> becomes a better approximation of <math>\pi(x)</math> as <math>x</math> grows larger. This also means that:

+

where <math>\pi(x)</math> is the number of primes up to and including <math>x</math>. This means that for a number <math>x</math>, the number of primes up to and including <math>x</math> approaches <math>x</math> divided by the the log to base e (or the natural log) of <math>x</math> and <math>{x/\ln(x)}</math> becomes a better approximation of <math>\pi(x)</math> as <math>x</math> grows larger. This also means that:

<math>P_x \approx x\ln(x)</math>, where <math>P_x</math> is the <math>x</math>th prime number.

<math>P_x \approx x\ln(x)</math>, where <math>P_x</math> is the <math>x</math>th prime number.

Revision as of 03:39, July 12, 2015

The Prime Number Theorem (or the PNT) is a theorem that concerns the distribution of primes and, subsequently, the gaps between primes. Its first proof date is not known.

Statement of Theorem

The theorem, formally stated, says that:

$ \lim_{x\to\infty}\frac{\pi(x)}{x/\ln(x)}=1 $

where $ \pi(x) $ is the number of primes up to and including $ x $. This means that for a number $ x $, the number of primes up to and including $ x $ approaches $ x $ divided by the the log to base e (or the natural log) of $ x $ and $ {x/\ln(x)} $ becomes a better approximation of $ \pi(x) $ as $ x $ grows larger. This also means that:

$ P_x \approx x\ln(x) $, where $ P_x $ is the $ x $th prime number.

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