(Created page with "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...") |
mNo edit summary Tag: sourceedit |
||
| (5 intermediate revisions by 2 users not shown) | |||
| Line 6: | Line 6: | ||
<math>\lim_{x\to\infty}\frac{\pi(x)}{x/\ln(x)}=1</math> |
<math>\lim_{x\to\infty}\frac{\pi(x)}{x/\ln(x)}=1</math> |
||
| − | where <math>\pi(x)</math> is the number of primes up to and including <math>x</math>. This means that for |
+ | 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. |
||
| + | |||
| + | {{Important Pages}} |
||
| + | [[Category:Important Pages]] |
||
| + | [[Category:Theorems and Proofs]] |
||
| + | [[Category:History of Prime Numbers]] |
||
Latest revision as of 07:08, 8 February 2017
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:
where is the number of primes up to and including . This means that for a number , the number of primes up to and including approaches divided by the the log to base e (or the natural log) of and becomes a better approximation of as grows larger. This also means that:
, where is the th prime number.
