The Least Number with Prescribed Legendre Symbols

TL;DR

This paper estimates how many integers up to a certain size can be represented by specific quadratic forms, using properties of Legendre symbols at multiple primes to understand their distribution.

Contribution

It introduces a novel approach to estimate the count of integers represented by quadratic forms based on Legendre symbol sign patterns at small primes.

Findings

Provides bounds on the number of representable integers up to X

Connects quadratic form representation to Legendre symbol sign vectors

Enhances understanding of integer representation by quadratic forms

Abstract

In this article we estimate the number of integers up to $X$ which can be represented by a positive-definite, binary integral quadratic form of discriminant which is small relative to $X$. This follows from understanding the vector of signs when computing the Legendre symbol of small integers $n$ at multiple primes.