The least quadratic non-residue

Abstract

For a prime number p p , we say a a is a quadratic non-residue modulo p p if there is no integer x x such that x 2 ≡ a mod p x^2\equiv a\bmod {p} . The problem of bounding the least quadratic non-residue modulo p p has a rich mathematical history. Moreover, there have been recent results, especially concerning explicit estimates. In this survey paper we give the history of the problem and explain many of the main achievements, giving explicit versions of these results in most cases. The paper is intended as a self-contained collection of the main ideas that have been used to attack the problem.