Quadratic Congruences. See attached file for full problem description.
(See attached file for full problem description) Let p be an odd prime. Prove that if p does not divide a and p does not divide b then . (Hint: Euler’s Criterion)
(See attached file for full problem description) Prove that if and q is an odd prime then .
(See attached file for full problem description) Suppose that is a prime. Prove that if a is a quadratic nonresidue modulo p, then a is a primitive root modulo p.
Use Quadratic Congruences to evaluate the following fractions. See attached file for full problem description.
(See attached file for full problem description) Determine which of the following quadratic congruences has solutions: a. b. c.
Classify every integer a, 1<=a<11 as to whether it is a quadratic residue or nonresidue modulo 11.
8. Find all solutions to the quadratic congruences, if they exist. (a) x2 + x + 1 ≡ 0 (mod 7). (b) x2 ≡ 55 (mod 179)
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.
Find a primitive root modulo 17 if it exists.
