The Least Number with Prescribed Legendre Symbols

2016/05/18 by Brandon Hanson, Hanson, Brandon, Robert C. Vaughan +4
Mathematics · #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #math.NT

paper · pdf · doi:10.48550/arxiv.1605.05584

This is the second edition of the article. An error from the first edition has been corrected

openalex publication_date 2016/05/18 · arxiv created 2016/10/29 · arxiv updated 2016/11/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Related

• A century problem related to the Legendre symbol modulo p
• When the number of divisors is a quadratic residue
• The least prime number represented by a binary quadratic form
• Legendre Symbol of ∏ f(i,j) over 0<i<j<p/2, p\nmid f(i,j)
• The Legendre Approximation and Arithmetic Bias