Almost universal sums of triangular numbers with one exception

TL;DR

This paper classifies all sums of triangular numbers that are almost universal with one exception and provides a criterion for such sums, extending the famous 15-theorem to this context.

Contribution

It offers a complete classification of almost universal sums of triangular numbers with one exception and generalizes the 15-theorem for these sums.

Findings

Classified all almost universal sums with one exception.

Developed an effective criterion for almost universality.

Extended the 15-theorem to sums of triangular numbers.

Abstract

For an arbitrary integer $x$, an integer of the form $T(x)=\frac{x^2+x}{2}$ is called a triangular number. For positive integers $\alpha_1,\alpha_2,\dots,\alpha_k$, a sum $\Delta_{\alpha_1,\alpha_2,\dots,\alpha_k}(x_1,x_2,\dots,x_k)=\alpha_1 T(x_1)+\alpha_2 T(x_2)+\cdots+\alpha_k T(x_k)$ of triangular numbers is said to be almost universal with one exception if the Diophantine equation $\Delta_{\alpha_1,\alpha_2,\dots,\alpha_k}(x_1,x_2,\dots,x_k)=n$ has an integer solution $(x_1,x_2,\dots,x_k)\in\mathbb{Z}^k$ for any nonnegative integer $n$ except a single one. In this article, we classify all almost universal sums of triangular numbers with one exception. Furthermore, we provide an effective criterion on almost universality with one exception of an arbitrary sum of triangular numbers, which is a generalization of "15-theorem" of Conway, Miller and Schneeberger.