On the least odd quadratic non-residue

Abstract

Given a prime p > 3 let νp be the least positive odd integer which is a non-square mod p. Then νp = q is a prime less than p, and one knows that \({\nu_p < \sqrt{p}}\) unless \({p \in \{5, 7, 11, 13, 23, 59, 109, 131\}}\) (Brauer, Nagell, Redei). Using analytical methods it has been shown that \({\nu_p = O(p^{\frac{1}{4\sqrt e}})}\) (Vinogradov, Burgess), and νp = O(log2p) if one assumes the Extended Riemann Hypothesis (Ankeny). In this paper we prove, with elementary algebraic means, that \({\nu_r < {\frac{3}{2}}{\rm log}{\frac{r-1}{2}}}\) for each prime \({r \equiv p\,({\rm mod} 4 \widehat{q})}\) , with at most one exception. Here \({\widehat{q}}\) is the product of all odd primes less than or equal to q.