Wonder whether any of you guys know why it is that the proof of the infinitude of primes that is based on the coprimality of any pair of (distinct) Fermat numbers is commonly attributed to Pólya.

In the first paragraph of this letter from Golbach to Euler there is already an argument along those lines, but since documents crediting it to Professor Pólya are not rare out there, it seems like it's passed unnoticed by a nonzero number of persons.

So, what do you think about this? It's not like Fermat numbers are essential to the proof or that there are no other demonstrations of the result... It's just that I'd really like to know about the origins of this discrepancy between the sources.