Vinogradov's Conjecture and Beyond

TL;DR

This paper establishes bounds on the occurrence of patterns of quadratic residues and nonresidues for large primes congruent to 3 mod 4, and relates this to the size of the least nonresidue, advancing understanding of Vinogradov's conjecture.

Contribution

It provides new upper bounds on pattern occurrences and the least nonresidue for large primes congruent to 3 mod 4, extending previous results in quadratic residue theory.

Findings

Bound on pattern occurrences as length tends to log base 2 of p

Existence of a constant c bounding the least nonresidue

Results hold for sufficiently large primes p ≡ 3 mod 4

Abstract

In this paper, if prime $p\equiv 3\pmod 4$ is sufficiently large then we prove an upper bound on the number of occurences of any arbitrary pattern of quadratic residues and nonresidues of length $k$ as $k$ tends to $\lceil \log_2 p\rceil$. As an immediate consequence, it proves that, there exist a constant $c$ such that, the least nonresidue for such primes is at most $c\lceil \log_2 p\rceil$.