Let P (x) = x(x 1) + 1 and x0 be any positive integer. Let xn+1 = P (xn), for n = 0, 1, 2, . . . . Prove that, for any n 2, the numbers x1, x2, . . . , xn are pairwise relatively prime. How does this provide an alternate proof of the fact that there are infintely many primes?