Euclid proved that there must be an infinite number of primes 23 centuries ago. His argument is based on a ruler-and-compass construction, using lengths of line segments. Today there at least 7 different ways to prove that there can be no largest prime, L. (There may exist dozens of proofs, I'm not sure what the total is.)

For now, lets look at this particular proof.

Assume there are not an infinite number of primes. Then there exists a largest prime, denote it L. Multiply all the primes in one large , but finite, list. Let m denote their product.

(2*3*5*7*11*13*17 * . . . * L) = m

Consider the number k = m+1 . Obviously L < k (by a large margin). What are the divisors of k?

For sport, we can make this proof smaller. Let L and P denote the two largest primes. Define q =L*P + 1 . What are the divisors of q?

This proof is so cute, that xckd made a haiku out of it. https://xkcd.com/622/