is there a largest prime?

To Prove: There is no largest prime.

Proof: By contradiction, let P = {2, 3, 5, 7, ..., P_L} be the finite set of primes, and let P_L be the largest prime. We construct a number

M = 1 + 2·3·5···P_L = 1 + ∏_i=1^L P_i

It follows that M is not evenly divisible by any prime, as M/P_i gives a remainder of 1 for all of the primes from 2 to P_L, inclusive. That is, M is prime and M > P_L, a contradiction, and our result is established.

QED

Note: The above proof is attributed to Euclid (circa 300 BC). Another proof, due to Euler, states that the sum of the reciprocals of all prime numbers diverges.

Source: wikipedia.org