Quote:
Originally Posted by xilman
I've not see the book itself so can't tell whether one of the six proofs of the infinitude of primes is the very elegant one based on the factorization of Mersenne numbers and Fermat numbers.
The basic idea is that F_n  2 = 2^2^n  1 = (2^2^(n1) +1) (2^2^(n1) 1) = F_{n1} * (F_{n1} 2)
by the difference of squares factorization formula and noting that F_n is coprime to F_m when m != m.

Yes, that's the 2nd one.
And the 3rd one uses Mersenne numbers (for prime p, a prime factor of \(M_p\) is greater than p).