What is Prime number theorem?
The '''prime number theorem''' is a [[theorem]] from [[number theory]]. [[Prime number]]s are not distributed evenly across the number range. The theorem formalizes the idea that the [[probability]] of hitting a prime number between 1 and a given number becomes smaller, as numbers grow. This probability is about n/ln(n), where ln(n) is the [[natural logarithm]] [[Function (mathematics)|function]]. This means that the probability of hitting a prime number with 2n [[digit]]s is about half as likely than with n digits. For example, among the positive integers of at most 1000 digits, about one in 2300 is prime (ln&nbsp;10<sup>1000</sup> ≈ 2302.6), whereas among positive integers of at most 2000 digits, about one in 4600 is prime (ln&nbsp;10<sup>2000</sup> ≈ 4605.2). In other words, the average gap between consecutive prime numbers among the first ''N'' integers is roughly ln(''N'').<ref>{{cite book|last = Hoffman|first = Paul|title = The Man Who Loved Only Numbers|url = https://archive.org/details/manwholovedonlyn00hoff_078|publisher = Hyperion|year = 1998|page = [https://archive.org/details/manwholovedonlyn00hoff_078/page/n251 227]|isbn = 0-7868-8406-1}}</ref> Fifteen-year old [[Carl Friedrich Gauss]] suspected that there was a link between prime numbers and [[logarithm]]s in 1793. [[Adrien-Marie Legendre]] also suspected such a link in 1798. [[Jacques Hadamard]] and [[Charles-Jean de La Vallée Poussin]] proved the prime number theorem in 1896, over a century after Gauss. ==References== {{reflist}} [[Category:Theorems in number theory]]