The smallest root of a polynomial congruence

Abstract

Fix f (t) ∈ Z[t] having degree at least 2 and no multiple roots. We prove that as k ranges over those integers for which the congruence f (t) ≡ 0 (mod k) is solvable, the least nonnegative solution is almost always smaller than k/(log k) c f . Here c f is a positive constant depending on f . The proof uses a method of Hooley originally devised to show that the roots of f are equidistributed modulo k as k varies.