Quadratic residues and applications (original) (raw)
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Number theory as a coherent mathematical subject started with the work of Fermat in the decade from 1630 to 1640, but modern number theory, that is, the systematic and mathematically rigorous development of the subject from fundamental properties of the integers, began in 1801 with the appearance of the landmark text of Gauss, Disquisitiones Arithmeticae. A major part of the Disquisitiones deals with quadratic residues and nonresidues. Beginning with these fundamental contributions of Gauss, the study of quadratic residues and nonresidues has subsequently led directly to many of the key ideas and techniques that are used everywhere in number theory today, and the primary goal of these lectures is to use this study as a window through which to view the development of some of those ideas and techniques. In pursuit of that goal, we will employ methods from elementary, analytic, and combinatorial number theory, as well as methods from the theory of algebraic numbers.
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In this note, we will present some olympiad problems which can be solved using quadratic congruences arguments. 1 Definitions and Properties Let x, y and z be integers, x > 1, y ≥ 1 and (x, z) = 1. We say that z is a residue of y − th degree modulo x if congruence n ≡ z (mod x) has an intenger solution. Otherwise z is a nonresidue of y− th degree. For x = 2, 3, 4 the residues are called quadratic, cubic, biquadratic, respectively. This article is mainly focused on quadratic residues and their properties. Lemma Let p be an odd prime. There are p−1 2 quadratic residues in the set {1, 2, 3..., p− 1}. 1.1 Legendre’s Symbol Given a prime number p and an integer a, Legendre’s symbol ( a p ) is defined as: ( a p ) = { 1 if a is a quadratic residue modulo p −1 otherwise (1) Property 1 If a ≡ b (mod p) and ab is not divisible by p, then (ap ) = ( b p ). Property 2 Legendre’s symbol is multiplicative, i.e. ( p ) = ( a p )( b p ) for all integers a, b and prime number p > 2. Property 3 I...
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Mathematics of computation, 2004
Let {s k , k ≥ 0} be the sequence defined from a given initial value, the seed, s 0 , by the recurrence s k+1 = s 2 k − 2, k ≥ 0. Then, for a suitable seed s 0 , the number M h,n = h • 2 n − 1 (where h < 2 n is odd) is prime iff s n−2 ≡ 0 mod M h,n. In general s 0 depends both on h and on n. We describe a slight modification of this test which determines primality of numbers h•2 n ±1 with a seed which depends only on h, provided h ≡ 0 mod 5. In particular, when h = 4 m − 1, m odd, we have a test with a single seed depending only on h, in contrast with the unmodified test, which, as proved by W. Bosma in Explicit primality criteria for h • 2 k ± 1, Math. Comp. 61 (1993), 97-109, needs infinitely many seeds. The proof of validity uses biquadratic reciprocity.
Quadratic residues and the combinatorics of sign multiplication
Journal of Number Theory, 2008
If S is a nonempty finite set of positive integers, we find a criterion both necessary and sufficient for S to satisfy the following condition: if q is a fixed nonnegative integer, then there exists infinitely many primes p such that S contains exactly q quadratic residues of p. This result simultaneously generalizes two previous results of the author, and the criterion used is expressed by means of a purely combinatorial condition on the prime factors of the elements of S of odd multiplicity.
Residue properties of certain quadratic units
Journal of Number Theory, 1985
Explicit formulas are given for the quadratic and quartic characters of units of certain quadratic fields in terms of representations by positive definite binary quadratic forms, as conjectured by Leonard and Williams (Pacific J. Math. 71 (1977), Rocky Mountain J. Math. 9 (1979)), and by Lehmer (J. Reine Angew. Math. 268/69 ). For example, if p and 4 are primes such that p = 1 (mod 8) 9 3 5 (mod 8) and the Legendre symbol (q/p) = 1, and if E is the fundamental unit of a(&), then (4~)~ = (-1 )"+T where p=a2+ 16b2 and pk= c2 + 16qd' with k odd. e