Numbers, Infinitely Many Primes

Infinitely Many Primes

Suppose there is a finite list of primes.  Multiply them together and add 1, giving n.  Now n is not prime - it is larger than all the primes on our list, which is suppose to be complete.  So n is composite.  Let p be a prime in the unique factorization of n.  Since p is on the list, it divides n-1, as well as n.  Hence it divides 1, which is impossible.  There are an infinite number of prime numbers.

The gaps between primes can be arbitrarily large.  Recall that n!, or n factorial, is the product of the first n integers.  Now n!+2 is divisible by 2, n!+3 is divisible by 3, and so on through n!+n, giving n-1 composite numbers in a row.

Primes that are only 2 apart are called twin primes.  Examples include 11 and 13, and 71 and 73.  We believe, but have not yet proved, that there are infinitely many twin primes.