Proof that there are infinitely many prime numbers

© Stu Savory, 2005.

A college reader from the US (whom I shall generously leave nameless) wrote saying he had read my various maths pages and liked in particular the pretty quilt of prime numbers I wrote about (Sieve of Eratosthenes page). Thanks. BUT!, he then asked What's the biggest prime?

There is no biggest prime, sir. There are an infinite number of primes. Remember, an integer is either itself prime or can be uniquely factored into a product of primes. So assume that there were a largest prime P. Now form the product of all primes upto and including prime P, write 2*3*5*7*11*13.....*P = N. Add one, giving M=N+1. Now M is either prime or factorable. If it is prime, it is bigger than P, thus leading to a contradiction. Assume it is factorable, then if you divide M by any of the primes 2 through P there will always be a remainder (1) left over. So M must have a factor larger than P, which is a contradiction again. Therefore there is no largest prime P. Quod erat demonstrandum!

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