Diagonal odd-regular ternary quadratic forms

TL;DR

This paper classifies diagonal odd-regular ternary quadratic forms, showing there are at most 147 such forms and confirming the odd-regularity of all but six candidates, advancing understanding of quadratic form representations.

Contribution

It provides a finite classification of diagonal odd-regular ternary quadratic forms and verifies their odd-regularity for nearly all candidates, which was previously unknown.

Findings

Maximum of 147 diagonal odd-regular ternary quadratic forms.

Odd-regularity confirmed for all but 6 candidates.

Progress in classifying quadratic forms based on local representation.

Abstract

A (positive definite primitive integral) quadratic form is called odd-regular if it represents every odd positive integer which is locally represented. In this paper, we show that there are at most 147 diagonal odd-regular ternary quadratic forms and prove the odd-regularities of all but 6 candidates.