# Regular ternary triangular forms

## TL;DR

This paper classifies all primitive regular ternary triangular forms, proving that exactly 49 such forms exist, which represent all locally represented positive integers.

## Contribution

It provides a complete classification of primitive regular ternary triangular forms, establishing the exact number as 49.

## Key findings

- Exactly 49 primitive regular ternary triangular forms exist.
- Each such form represents all positive integers locally represented.
- The classification completes the understanding of regular ternary triangular forms.

## Abstract

An integer of the form $T_x=\frac{x(x+1)}2$ for some positive integer $x$ is called a triangular number. A ternary triangular form $aT_{x}+bT_{y}+cT_{z}$ for positive integers $a,b$ and $c$ is called regular if it represents every positive integer that is locally represented. In this article, we prove that there are exactly 49 primitive regular ternary triangular forms.