Changes

The number of primes smaller than a given number <math>N</math> is roughly <math>\frac{N}{\ln(N)}</math>, where <math>\ln(N)</math> is the [[natural logarithm]] (base ''[[e]]'') of <math>N</math>.<ref>http://home.att.net/~numericana/answer/primes.htm</ref>  This is a formulation of a more general statement known as the [[prime number theorem]].

The number of primes smaller than a given number <math>N</math> is roughly <math>\frac{N}{\ln(N)}</math>, where <math>\ln(N)</math> is the [[natural logarithm]] (base ''[[e]]'') of <math>N</math>.<ref>http://home.att.net/~numericana/answer/primes.htm</ref>  This is a formulation of a more general statement known as the [[prime number theorem]].

==The Prime Numbers==

==How many prime numbers==

{{main|prime number theorem}}

{{main|prime number theorem}}

It is easy to prove that there are an infinite number of primes using Euclid's second theorem:  Imagine there is a finite set consisting of all the primes.  Multiply them all together, add 1, and call this N.  N would not be divisible by any number in the set--there would always be a remainder of 1.  Because all non-prime numbers can be decomposed into a product of underlying primes, N must be divisible by some prime not in the set (possibly itself), thus contradicting the assumption that the set contained all of the primes.

It is easy to prove that there are an infinite number of primes using Euclid's second theorem:  Imagine there is a finite set consisting of all the primes.  Multiply them all together, add 1, and call this N.  N would not be divisible by any number in the set--there would always be a remainder of 1.  Because all non-prime numbers can be decomposed into a product of underlying primes, N must be divisible by some prime not in the set (possibly itself), thus contradicting the assumption that the set contained all of the primes.

Another proof that there are infinitely many primes shows something stronger, namely the sum of the reciprocals of primes less than n "grows like" log(log(n)):

Another proof that there are infinitely many primes shows something stronger, namely the sum of the reciprocals of primes less than n "grows like" log(log(n)):

<math>\sum_{p\le x}p^{-1}\approx\log\log x</math>

<math>\sum_{p\le x}p^{-1}\approx\log\log x</math>

==Alternative Definition==

==Alternative Definition==