is there a largest prime?

To Prove: There is no largest prime.

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

M = 1 + 2·3·5···PL = 1 + ∏i=1L Pi

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


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.


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