The Least Quadratic Non Residue

Abstract

This, the problem of the least quadratic non residue, has often been investigated. The best result is due to Vinogradov, who proved that (1) n(k) = O(k1I(2Ve) where n(k) denotes the least positive quadratic non residue of the prime k. Also, several interesting results about n(k) have been obtained by A. Brauer through elementary methods. It has been conjectured that n(k) ko(E). This was first proved by Linnik on the assumption of the extended Riemann Hypothesis (E.R.H.). Recently, with the same hypothesis, S. Chowla and P. Erdds have improved Linnik's result to n(k) = O(exp ((log k)i+e)) In this paper I prove on the basis of the E.R.H. (2) n(k) = O((log k)2) S. Chow-la has proved that there exist infinitely many primes k where the first cl log k residues (mod k) are all quadratic residues. Hence, the upper bound on (2) cannot be improved beyond O(log k). The two results, the upper and lower bounds, are now not too far apart. The best possible bound for n(k) is probably not the bound in (2). One COuld(1 improve the bound in (2) by methods in this paper if one could derive any non trivial estimate in terms of k on