1) If k > 2, and n = 2^{k-3}, show that 3^{n}is not congruent to 1 (mod 2^{k}).

2) Let p > 3 be a prime, and suppose that p º 3 (mod 4). If q = 2p+1 is also prime show that 2^{p}-1 is never prime.

3) Let p be a prime ¹ 2 or 5, and let N be any integer. If p | (N^{2}-5) show that p is of the form 1+5k, or 4+5k

4) Let p be a prime. Show that every prime divisor of 2^{p }-1 is > p.