The Least Quadratic Non Residue

Abstract

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.

Certain hypotheses concerningL-functions

“…In the case of the Extended Riemann Hypothesis (ψ = 1/2 log q), we have the slightly stronger result of Ankeny [1] Y-< log 2 q and a similar result for Y + .…”

Searching for Primitive Roots in Finite Fields

“…This was first proved by Ankeny [2]. The following simple variant of Ankeny's theorem will be useful.…”

“…Our first result applies to the problem searching for primitive roots in GF(2"), and more generally, in GF(pn) with p small. We show (unconditionally) that given any irreducible polynomial / of degree n in GF (2)[X], there exists a polynomial 6 e GF (2)[Ar] (itself irreducible) such that degö = 0(log«) and (dmodf) is a primitive root for GF (2)[X]/(/) = GF(2"). More precisely, we prove the following: Theorem 1.1.…”

A generalization of Miller’s primality theorem

“…Miller proved his result by using a result of Ankeny [2] on the least quadratic nonresidue. Ankeny proved that under ERH the least quadratic nonresidue modulo a prime p is O((log p) 2 ).…”

“…The main theorem in [5] is the following: Suppose n ∈ A r is not a prime, n = (1 + r) 2 , and ω as above; then n is an ω-prime to base a for at most…”