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Criterion for r th power residuacity

1960; Mathematical Sciences Publishers; Volume: 10; Issue: 4 Linguagem: Inglês

10.2140/pjm.1960.10.1115

1945-5844

N. C. Ankeny,

The Law of Quadratic Reciprocity in the rational integers states: If p, q are two distinct odd primes, then q is a square (modp) if and only if (l) {p -1)l2 p is a square (modg).One of the classical generalizations of the law of reciprocity is of the following type.Let r be a fixed positive integer, φ(r) denotes the number of positive integers <£ r which are relatively prime to r; p, q are two distinct primes and p == 1 (mod r).Then can we find rational integers a λ (p) f a 2 (p),, a h {p) determined by p, such that q is an rth power (modp) if and only if ajjή, •• ,α /i (;p) satisfy certain conditions (mod q).The Law of Quadratic Reciprocity states that for r = 2, we may take a λ {p) = (-l) (2J ~1)/2 p.Jacobi and Gauss solved this problem for r = 3 and r = 4, respectively.Mrs. E. Lehmer gave another solution recently [2].In this paper I would like to develop the theory when r is a prime and q = 1 (modr).I then show that q is an rth power (modp) if and only if a certain linear combination of a λ {p), , α r -i(p) is an rth power (mod q). a 1 (p) f , α Γ _ x (p) are determined by solving several simultaneous Diophantine equations.This determination appears mildly formidable and to make the actual numerical computations would certainly be so for a large r.(See Theorem B below.)Also given is a criterion for when r is an rth power (mod p) in terms of a linear combination of Gi(p), * > α r-i(p) (modr 2 ).(See Theorem A below.)It is possible by the methods developed in this paper to eliminate the conditions that r is a prime and q = 1 (mod r).This would complicate the paper a great deal, and the cases given clearly indicate the underlying theory.Consider the following Diophantine equations in the rational integers: r Σ 5=1 ~ ( § ^)2 = (r -M=l / V (1where Xl fc) denotes the sum over all j lf , j k+1 -1, 2, , r -1, with the condition j x + + j kkj k+1 = i (mod r).