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.

Recoverable Formal Language

“…Patterns formed by quadratic residues and non-residues modulo a prime have been studied since the 19th century [21] and still they continue to attract attention of contemporary mathematicians [23,27,22,20] from various points of view. For a detailed historical overview of the concept of quadratic residues, we refer the reader to the monograph [24], and to the modern analysis from an algebraic-geometric point of view -to [20].…”

Upper Bound Of Least Quadratic Nonresidues

“…A recent proof appears in [13,Theorem 2.4], and the earliest proof in [17]. The generalization to arbitrary characters χ = 1 modulo p, and different approaches to the proofs are also available in the literature.…”

“…where γ > 0 is Euler constant. Substituting the parameter z = x 1/e , simplifying it, and comparing it with the assumed estimate for the character sum in (13), return…”