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.