Let р be a prime such that p = 22" + 1, for some n € N, with n > 1. (a) Determine if 5 is a quadratic residue or a quadratic nonresidue modulo p. (b) Use the result of (a) to prove that 5 is a primitive root modulo p. (c) Use the result of (b) to determine a complete set of representatives for all the solutions modulo 257 of x6 = 25(mod 257).
Let р be a prime such that p = 22" + 1, for some n € N, with n > 1. (a) Determine if 5 is a quadratic residue or a quadratic nonresidue modulo p. (b) Use the result of (a) to prove that 5 is a primitive root modulo p. (c) Use the result of (b) to determine a complete set of representatives for all the solutions modulo 257 of x6 = 25(mod 257).
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter2: The Integers
Section2.5: Congruence Of Integers
Problem 37E
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Transcribed Image Text:be a prime such that p = 2²″ + 1, for some n € N, with n > 1.
Let p
(a) Determine if 5 is a quadratic residue or a quadratic nonresidue modulo p.
(b) Use the result of (a) to prove that 5 is a primitive root modulo p.
(c) Use the result of (b) to determine a complete set of representatives for all
the solutions modulo 257 of x6 = 25(mod 257).
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