On ternary quadratic forms over the rational numbers

TL;DR

This paper provides an elementary proof that positive definite ternary quadratic forms over rationals do not represent all positive integers, and constructs specific classes of integers not represented by such forms.

Contribution

It offers a simple proof using quadratic reciprocity to show limitations of ternary quadratic forms in representing positive integers.

Findings

Positive definite ternary forms fail to represent infinitely many positive integers.

Constructs specific congruence classes not represented by any ternary form.

Minimum variables needed for universal positive integer representation is four.

Abstract

In this note, we give an elementary proof of the following classical fact. Any positive definite ternary quadratic form over the rational numbers fails to represent infinitely many positive integers. For any ternary quadratic form (positive definite or indefinite), our method constructs certain congruence classes whose elements, up to a square factor, are the only elements not represented over the rational numbers by that form. In the case of a positive definite ternary form, we show that these classes are non-empty. This shows that the minimum number of variables in a positive definite quadratic form representing all positive integers is four. Our proof is very elementary and only uses quadratic reciprocity of Gauss.