Even universal sums of triangular numbers
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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 $\alpha_1,\ldots,\alpha_k$, a sum $\alpha_1 T(x_1)+\cdots+\alpha_k T(x_k)$ of triangular numbers is said to be even universal if the diophantine equation $\alpha_1 T(x_1)+\cdots+\alpha_k T(x_k)=2n$ has an integer solution $(x_1,\ldots,x_k)\in\mathbb{Z}^k$ 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.
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 $\alpha_1,\ldots,\alpha_k$, a sum $\alpha_1 T(x_1)+\cdots+\alpha_k T(x_k)$ of triangular numbers is said to be even universal if the diophantine equation $\alpha_1 T(x_1)+\cdots+\alpha_k T(x_k)=2n$ has an integer solution $(x_1,\ldots,x_k)\in\mathbb{Z}^k$ 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.