Please help my, Thanks
Carefully explain you answers!
(1) Let n = 3^(t-1). Show that 2^n = -1 (mod 3^t). (Hint: 2 is a primitive root mod 3^2.)
(2) a) Let n be an integer >1, and suppose that p = 2^n+1 is a prime. Show that 3^((p-1)/2) +1
is divisible by p. (Hint: First show that n must be even.)
b) If p = 2^n+1, n>1, and 3^((p-1)/2) = -1 (mod p) show that p is a prime.
(3)If n is positive integer what is the number of solutions (x,y) (with x and y positive
integers) to the equation
1/x + 1/y = 1/n .
Carefully explain your reasoning.
(5) Let p be a prime. Show that every prime divisor of 2^p -1 is > p.