On the least square-free primitive root modulo p

Abstract

Let g (p) denote the least square-free primitive root modulo p. We show that g (p) < p 0.96 for all p.

Explicit upper bounds on the least primitive root

“…A quick computational check establishes this inequality for all X ≥ 2. This establishes (5) and proves the proposition.…”

“…Cohen, Oliveira e Silva and Trudgian [4] proved that g(p) ≤ 5.2p 0.99 . This was improved to p 0.96 by Cohen and Trudgian [5], to p 0.88 by Hunter [8], and to p 0.68 by Pretorius [16]. These latter results use numerically efficient versions of the Pólya-Vinogradov inequality (see, e.g., [6]) together with some amount of computation.…”

The least primitive root modulo $p^{2}$

“…While explicit upper bounds on g(p) have been given [4,5,10,15], we are unaware of any such bounds for h(p). Trivially, we have h(p) < p 2 , and, if we use an appropriate version of the Pólya-Vinogradov inequality [7], we can make the estimate h(p) < p 1+ǫ explicit.…”

Square-full primitive roots

“…In the absence of such an improvement many authors have proved the existence of small primitive roots with additional properties. For example Ha [7] has shown that for all primes p there exists a prime primitive root ≪ p 3.1 ; Cohen and Trudgian [4] showed that for all primes p there is a square-free primitive root less than p 0.96 -this was improved by Hunter [9] to p 0.88 . In this article we consider square-full primitive roots.…”