Even universal 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. 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.

arXiv comment12 pages. arXiv admin note: text overlap with arXiv:2201.04355