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Ifa andb are both quadratic residues of the odd primepor both nonresidues, show that the congruenceax2 ≡b (mod p) has a solution

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HOMEWORK 6

MATH-UA 0248-001 THEORY OF NUMBERS due on Nov, 2, 2020

1. Determine all the primitive roots of the prime 17.

2. If ab ≡ r (mod p), where r is a quadratic residue of the odd prime p, prove that a and b are both quadratic residues of p or both nonresidues mod p.

3. Ifa andb are both quadratic residues of the odd primepor both nonresidues, show that the congruenceax2 ≡b (mod p) has a solution. (Hint: multiply by a0 with aa0 ≡1 mod p).

4. Evaluate the following Legendre symbols:

(a) (7173);

(b) (127033658 )(Hint: 3658 = 2·31·39and 12703 is prime).

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