## 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