1. ## Modular arithmetic

(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 .

(5) Let p be a prime. Show that every prime divisor of 2^p -1 is > p.

2. See here, here and here

3. ## thank you

thank you very much, for Your halp.

It's not for me (I'm astronomer).