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4. For an odd prime p, suppose that (=) = 1 and consider the congruence P 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 = ±ak+¹, or x = ±2²k+1k+1 a 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 P 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 = ±ak+¹, or x = ±2²k+1k+1 a 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

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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[Number Theory] How do you solve question 4? 

1. Evaluate the following Legendre symbols: 2. Use Gauss' Lemma, (i) to compute (ii) (iii) In (ii) p is an odd prime, p ‡ 7; in (iii) p is an odd prime p ‡ 3. In (ii) express the answer in terms of a congruence condition on p mod 28, and in (iii) express the answer in terms of a congruence condition on p mod 24, analogous to = 1 if p = 1 mod 4 and -1 if p = 3 mod 4. 463 541 = (-1)", 4. For an odd prime p, suppose that 31 8933 104729 3. For an odd prime p≥ 7, show that there are consecutive quadratic residues mod p, i.e., that there exists an element a € Up such that μ = |aPN, a (²) - ( ² + ¹) - = Hint: First show that at least one of the numbers 2, 5 and 10 is a quadratic residue mod p. x = a = 1 and consider the congruence = 1. = x² = a mod p. (i) For p = 3 + 4k, i.e., for p = 3 mod 4, show that the solutions to the congruence are tak+1 or x = 1 (ii) For p = 5 + 8k, i.e., for p = 5 mod 8, show that the solutions to the congruence are x = ±ak+¹, +2²k+1 k+1 a 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
Transcribed Image Text:1. Evaluate the following Legendre symbols: 2. Use Gauss' Lemma, (i) to compute (ii) (iii) In (ii) p is an odd prime, p ‡ 7; in (iii) p is an odd prime p ‡ 3. In (ii) express the answer in terms of a congruence condition on p mod 28, and in (iii) express the answer in terms of a congruence condition on p mod 24, analogous to = 1 if p = 1 mod 4 and -1 if p = 3 mod 4. 463 541 = (-1)", 4. For an odd prime p, suppose that 31 8933 104729 3. For an odd prime p≥ 7, show that there are consecutive quadratic residues mod p, i.e., that there exists an element a € Up such that μ = |aPN, a (²) - ( ² + ¹) - = Hint: First show that at least one of the numbers 2, 5 and 10 is a quadratic residue mod p. x = a = 1 and consider the congruence = 1. = x² = a mod p. (i) For p = 3 + 4k, i.e., for p = 3 mod 4, show that the solutions to the congruence are tak+1 or x = 1 (ii) For p = 5 + 8k, i.e., for p = 5 mod 8, show that the solutions to the congruence are x = ±ak+¹, +2²k+1 k+1 a 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
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