Bounds for pairs of consecutive seventh and higher power residues
John D. Brillhart, Derrick Henry Lehmer, Emma T. Lehmer · Mathematics of Computation · 1964
Introduction.In previous papers [1], [3], [4] and [5] the first occurrence of two consecutive kth power residues of a prime p was discussed for k = 2,3, 4, 5, and 6.The present paper is concerned with the same problem for A; > 6.Before giving the results of this investigation in detail we need to recall some definitions and notations.Let fc be an integer > 1, and let p = km + 1 be a prime.Let g be a primitive root of p and let oin * = x (mod p).Let Rin) = ind n (mod k), 0 ^ 7¿(w) < fc.[In particular n is a fcth power residue if and only if Rin) = 0.] Let r = r(fc, p) denote the least positive r such that r and r + 1 are both fcth powers modulo p, so that ñ(r) = fi(r +1) =0.Let a prime p* = p*(fc) for which no r exists be called an "exceptional prime."Let A(fc) = max r(fc, p) taken over all nonexceptional primes p.Let any vector whose components are non-negative integers less than fc be called a "case vector."For fc < 8, to each case vector [ci, c2, ■ • ■ , ct] there corresponds an infinite class of primes p for whichwhere qt is the tth prime [6].A case vector [ñ(2), ¿2(3), • • • , Riqt)] characterizing an infinite class of primes for which r(fc, p) = A(fc) will be called a "maximal case vector." 1. Results.For convenience of reference new and old results are given in Table I.Previous results are recognized by references in square brackets.