On the representation of integers by quadratic forms

TL;DR

This paper investigates the representation of integers by quadratic forms, offering new bounds for solutions in indefinite cases and improved bounds for insolubility in positive definite cases, advancing understanding in number theory.

Contribution

It provides novel upper bounds for solutions of indefinite quadratic forms and enhances bounds for insoluble integers in positive definite forms.

Findings

New upper bounds for least solutions when Q is indefinite.

Improved bounds for integers k where Q=k is locally soluble but globally insoluble.

Advances in understanding the representation of integers by quadratic forms.

Abstract

Let Q be a non-singular quadratic form with integer coefficients. When Q is indefinite we provide new upper bounds for the least non-trivial integral solution to the equation Q=0. When Q is positive definite we provide improved upper bounds for the least positive integer k such that the equation Q=k is insoluble in integers, despite being soluble modulo every prime power.