For any n ≥ 0, let Fn = 22"n+1. (These Fn are often called Fermat numbers. Those that...

For any n ≥ 0, let Fn = 22"n+1. (These Fn are often called Fermat numbers. Those that are prime are the Fermat primes introduced in Section 4.3.)

(a) Prove that for all n ≥ 1.

(b) Prove that gcd(Fm, Fn) = 1 for any positive integers m and n with m ≠ n.

(c) Use the result of (b) to give another proof (different from Euclid’s) that there are infinitely many primes.