LetFor example, F1 = 5, F2 = 17,F3 = 257, and F4 = 65537. Fermat thought that all the Fk’s...

LetFor example, F1 = 5, F2 = 17,F3 = 257, and F4 = 65537. Fermat thought that all the Fk’s might be prime, but Euler showed in 1732 that F5 factors as 641 • 6700417, and in 1880 Landry showed that F6 is composite. Primes of the form Fk are called Fermat primes. Show that if k≠m, then the numbers Fk and Fm have no common factors; that is, show that gcd(Fk, Fm) = 1. [Hint. If k > m, show that Fm divides Fk − 2.]