For an alternative to Euclid’s proof, there is Furstenberg’s topological proof of the infinitude of primes. The MAA‘s Mathematics Magazine published a variation of Furstenberg’s proof which does not use topology:

For each prime *p*, let *S _{p}* =

*p*

**Z**. Each

*S*is periodic (it’s characteristic function is periodic). Let

_{p}*S*be the union of all

*S*. If

_{p}*S*is the union of finitely many periodic sets, then

*S*is also periodic. However, the complement of

*S*is {-1, 1}, so

*S*is not periodic. Hence, there must be infinitely many sets

*S*, and infinitely many primes.

_{p}