On the number of nonquadratic residues which are not primitive roots

Abstract

Abstract. We show that there exist infinitely many positive integers r not of the form (p − 1)/2 − φ(p − 1), thus providing an affirmative answer to a question of Neville Robbins.For every positive integer n let φ(n) be the Euler function of n. For an odd prime number p put f (p) = (p − 1)/2 − φ(p − 1). Note that f (p) counts the number of quadratic nonresidues modulo p which are not primitive roots. [91]

Artin's Primitive Root Conjecture – A Survey

“…Modulo p the number of nonquadratic residues which are not primitive roots is obviously g(p) := (p − 1)/2 − ϕ(p − 1). The values assumed by g were investigated by Luca and Walsh [316] and by Robbins [430]. More recently, Gun et al [199] applied character sum estimates to prove results on consecutive quadratic non-residues modulo p that are not primitive roots.…”

Nonaliquots and Robbins numbers

“…At the 2002 Western Number Theory Conference in San Francisco, Neville Robbins asked whether there exist infinitely many positive integers m for which f r (p) = m has no solution; let us refer to such integers as Robbins numbers. The existence of infinitely many Robbins numbers has been shown recently by Luca and Walsh [4], who proved that for every odd integer w ≥ 3, there exist infinitely many integers ℓ ≥ 1 such that 2 ℓ w is a Robbins number. In Theorem 2 (Section 3), we show that the set of Robbins numbers has a positive density; more precisely, if…”

When the Sum of Aliquots Divides the Totient

“…and let s(n) = σ(n) − n be the sum of the aliquot divisors of n 1. The function s(n) and related arithmetic functions (such as f (n) = n − ϕ(n)) have been previously studied in the literature (see, for example, [1,2,[5][6][7][8]).…”