Even universal sums of triangular numbers

TL;DR

This paper classifies all sums of scaled triangular numbers that can represent every even nonnegative integer and provides a criterion to determine their universality, extending the classical triangular theorem of eight.

Contribution

It offers a complete classification of even universal sums of triangular numbers and generalizes the triangular theorem of eight with an effective criterion.

Findings

All even universal sums of triangular numbers are classified.

An effective criterion for checking even universality is established.

The results extend the classical triangular theorem of eight.

Abstract

For an arbitrary integer $x$, an integer of the form $T(x)\!=\!\frac{x^2+x}{2}$ is called a triangular number. Let $\alpha_1,\dots,\alpha_k$ be positive integers. A sum $\Delta_{\alpha_1,\dots,\alpha_k}(x_1,\dots,x_k)=\alpha_1 T(x_1)+\cdots+\alpha_k T(x_k)$ of triangular numbers is said to be even universal if the Diophantine equation $\Delta_{\alpha_1,\dots,\alpha_k}(x_1,\dots,x_k)=2n$ has an integer solution $(x_1,\dots,x_k)\in\mathbb{Z}^k$ for any nonnegative integer $n$. In this article, we classify all even universal sums of triangular numbers. Furthermore, we provide an effective criterion on even universality of an arbitrary sum of triangular numbers, which is a generalization of the triangular theorem of eight of Bosma and Kane.