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Unformatted text preview: od 7) 32 = 9 a 2 (mod 7) 42 = 16 a 2 (mod 7) 52 = 25 a 4 (mod 7) 62 = 36 a 1 (mod 7) Note that each quadratic residue appears twice on this list. There are no values of x which satisfy any of these equations: x2 a 3 (mod 7) x2 a 5 (mod 7) x2 a 6 (mod 7) The quadratic nonresidues modulo 7, the numbers that are not quadratic residues, are 3, 5, and 6. Although I will not do so here, it is easy to prove that, when p is odd, there are exactly (p1)/2 quadratic residues mod p and the same number of quadratic nonresidues mod p. Also, if a is a quadratic residue mod p, then a has exactly two square roots, one of them between 0 and (p- 1)/2, and the other between (p- 1)/2 and (p- 1). One of these square roots is also a quadratic residue mod p; this is called the principal square root. If n is the product of two primes, p and q, there are exactly (p- 1)(q- 1)/4 quadratic residues mod n. A quadratic residue mod n is a perfect square modulo n. This is because to be a square mod n, the residue must be a square mod p and a square mod q. For example, there are 11 quadratic residues mod 35: 1, 4, 9, 11, 14, 15, 16, 21, 25, 29, and 30. Each quadratic residue has exactly four square roots. Legendre Symbol
The Legendre symbol, written L(a,p), is defined when a is any integer and p is a prime greater than 2. It is equal to 0, 1, or -1. L(a,p) = 0 if a is divisible by p. L(a,p) = 1 if a is a quadratic residue mod p. L(a,p) = - 1 is a is a quadratic nonresidue mod p. One way to calculate L(a,p) is: L(a,p) = a(p- 1)/2 mod p Or you can use the following algorithm: 1. If a = 1, then L(a,p) = 1 2. If a is even, then L(a,p) = L(a /2,p)*(- 1)(p2- 1)/8 3. If a is odd (and ` 1), then L(a,p) = L(p mod a,a)*(- 1)(a- 1)*(p- 1)/4 Note that this is also an efficient way to determine whether a is a quadratic residue mod p (when p is prime). Previous Table of Contents Next Products | Contact Us | About Us | Privacy | Ad Info | Home Use of this site is subject to certain Terms & Conditions, Copyright © 1996-2000 EarthWeb Inc. All rights reserved. Reproduction whole or in part in any form or medium without express written permission of EarthWeb is p...
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This note was uploaded on 10/18/2010 for the course MATH CS 301 taught by Professor Aliulger during the Fall '10 term at Koç University.
- Fall '10