Quadratic congruences

8. Find all solutions to the quadratic congruences, if they exist. (a) x2 + x + 1 ≡ 0 (mod 7). (b) x2 ≡ 55 (mod 179)

Primitive numbers

Assume that n is odd and a is a primitive root mod n. Let b be an integer with b ≡ a(mod n) and gcd (b, 2n) =1. Show that b is a primitive root mod 2n.

Primitive root

Find a primitive root modulo 17 if it exists.

Primitive roots

Determine which elements of Z_7 (Z sub 7) are primitive roots.

Theory of Numbers (I): Principle of Mathematical Induction: Prove that 1^2 + 2^2 + 3^2 + … +n^2 = n(n+1)(2n + 1) / 6

Theory of Numbers (I) Principle of Mathematical Induction Prove that 1^2 + 2^2 + 3^2 + … +n^2 = n(n+1)(2n + 1) / 6