Representation of Integers by Ternary Quadratic Forms: A Geometric Approach

TL;DR

This paper extends geometric methods to determine which integers are represented by specific ternary quadratic forms, providing exact characterizations and sufficient conditions for representation.

Contribution

It generalizes Ankeny's geometric approach to new quadratic forms, offering precise criteria for integer representation by these forms.

Findings

Exact characterization of integers represented by x^2 + 2y^2 + 2z^2 and x^2 + y^2 + 2z^2

Sufficient conditions for representation by x^2 + y^2 + 3z^2 and x^2 + y^2 + 7z^2

Extension of geometric techniques to broader classes of quadratic forms

Abstract

In 1957 N.C. Ankeny provided a new proof of the three squares theorem using geometry of numbers. This paper generalizes Ankeny's technique, proving exactly which integers are represented by $x^2 + 2y^2 + 2z^2$ and $x^2 + y^2 + 2z^2$ as well as proving sufficient conditions for an integer to be represented by $x^2+y^2+3z^2$ and $x^2 + y^2 + 7z^2$.