#### How many primes are there?

It is true that the bigger a number is the more likely a smaller number divides it (making it composite) and although it is true to say that primes become more scarce, there is no end to them.

The proof is based on a powerful concept developed by Euclid of Alexandria (circa 3rd century BCE) called proof by contradiction. We begin by taking our idea (that primes go on forever) and assume the opposite is true (that there is a last prime). By building on this assumption we work towards an unavoidable contradiction. From this we conclude that if our opposite idea is wrong, then the original idea must be true.

Begin by assuming that the list of primes is finite: p1, p2, p3, ... , pn, where pn is the "last" prime.

Let N = p1× p2× p3× ... × pn + 1.

Clearly N is either composite or prime, but as N is bigger than any of the primes p1, p2, p3, ... , pn, it cannot be prime and must be composite.

But as p1× p2× p3× ... × pn is divisible by each of the known primes, N will not divide by any of the primes; it will always have a remainder of one.

So there must exist a prime factor bigger than the "last" prime, pn, which divides N. Hence our assumption, that the number of primes is finite (has an end), was wrong, and we conclude that the list of primes is without bound and is infinite.