4. For an odd prime p, suppose that (=) = 1 and consider the congruence x² = a mod p. (i) For p = 3 + 4k, i.e., for p = 3 mod 4, show that the solutions to the congruence are x = ±ak+1. (ii) For p = 5 + 8k, i.e., for p = 5 mod 8, show that the solutions to the congruence are x = tak+1, or x = ±2²k+1k+1 1 Explain how to see which is the correct choice. (iii) Use the results of (i) and (ii) to find the solutions to the congruences x² = 23 mod 751 x² = 15 mod 797
4. For an odd prime p, suppose that (=) = 1 and consider the congruence x² = a mod p. (i) For p = 3 + 4k, i.e., for p = 3 mod 4, show that the solutions to the congruence are x = ±ak+1. (ii) For p = 5 + 8k, i.e., for p = 5 mod 8, show that the solutions to the congruence are x = tak+1, or x = ±2²k+1k+1 1 Explain how to see which is the correct choice. (iii) Use the results of (i) and (ii) to find the solutions to the congruences x² = 23 mod 751 x² = 15 mod 797
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 58E: a. Prove that 10n(1)n(mod11) for every positive integer n. b. Prove that a positive integer z is...
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