Non primitive roots with a prescribed residue pattern

Abstract

Let p be an odd prime. If an integer g generates a subgroup of index t in (Z/ pZ) ∗, then we say that g is a t-near primitive root modulo p. In this paper, for a subset { a<inf>1</inf>, a<inf>2</inf>, ⋯ , a<inf>n</inf>} of Z\ { - 1 , 0 , 1 } , we prove each coprime residue class contains a positive density of primes p not having a<inf>i</inf> as a t-near primitive root and with the a<inf>i</inf> satisfying a prescribed residue pattern modulo p, for 1 ≤ i≤ n. We also prove a more refined variant of it. © 2023, Indian Academy of Sciences.