Let p = 2^k + 1 be a prime for some k ≥ 1. Let a be any integer such that gcd(a, p) = 1 and a is NOT a square mod p. Show that a is a primitive root mod p

Show that if p = 1 (mod 4), then x² = -1 (mod p) has a solution given by the least residue (mod p) of ((p-1)/2)!.

All integers prove n> 1, and x = y (mod n), then x^2 = y^2 (mod n)

Find all three incongruent solutions to the congruence x³ = 1 (mod 103). Express these solutions as a power of 5. (Hint: use the fact that 5 is the least positive primitive root modulo 103 and the theorem stating that if the integer a has order k modulo m and h> 0, then a" has order modulo m. k (h,k)