Euclid's proof of the infinitude of primes
This is proved by contradiction. Suppose there is a finite number of primes and let them be . Let
. Then we have
. When divided by any of the primes
,
leaves a remainder of 1 implying that either
is prime or that it has some other prime factors not in the set
. In any case we have it so that
does not contain all prime numbers. Contradiction!