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The smallest prime numbers are 2, 3, 5, 7, 11, 13...

The smallest prime numbers are 2, 3, 5, 7, 11, 13...

It is easy to prove that there are an infinite number of primes using [[Euclid's second theorem]]. If there were a finite number of primes, you could multiply them all together and add 1.  The resulting number would show the existence of a new prime, since it would not be divisible by any smaller prime (it would always have a remainder of 1).

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.

To construct a table of all prime numbers less than ''n'', you would use a method called the [[Sieve of Eratosthenes]].  Simply write down an ordered list of all counting numbers greater than 1.  Beginning with 2*2 = 4, cross out every second number:

To construct a table of all prime numbers less than ''n'', you would use a method called the [[Sieve of Eratosthenes]].  Simply write down an ordered list of all counting numbers greater than 1.  Beginning with 2*2 = 4, cross out every second number: