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, Let p be an odd prime. Prove or disprove the following statements. (a) If p = 2" + 1, then every quadratic nonresidue of p is a primitive root of p. (b) If a € Z with gcd(a, p) = 1, p = 3 (mod 8), and a is a quadratic residue of p, then p - 2a is a quadratic nonresidue of p.

, Let p be an odd prime. Prove or disprove the following statements. (a) If p = 2" + 1, then every quadratic nonresidue of p is a primitive root of p. (b) If a € Z with gcd(a, p) = 1, p = 3 (mod 8), and a is a quadratic residue of p, then p - 2a is a quadratic nonresidue of p.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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, Let p be an odd prime. Prove or disprove the following statements. (a) If p = 2" + 1, then every quadratic nonresidue of p is a primitive root of p. (b) If a € Z with gcd(a, p) = 1, p = 3 (mod 8), and a is a quadratic residue of p, then p - 2a is a quadratic nonresidue of p.
Transcribed Image Text:, Let p be an odd prime. Prove or disprove the following statements. (a) If p = 2" + 1, then every quadratic nonresidue of p is a primitive root of p. (b) If a € Z with gcd(a, p) = 1, p = 3 (mod 8), and a is a quadratic residue of p, then p - 2a is a quadratic nonresidue of p.
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