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arXiv · 0903.3026

Representing Sets with Sums of Triangular Numbers

Ben Kane
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Abstract

We investigate here sums of triangular numbers $f(x):=\sum_i b_i T_{x_i}$ where $T_n$ is the $n$-th triangular number. We show that for a set of positive integers $S$ there is a finite subset $S_0$ such that $f$ represents $S$ if and only if $f$ represents $S_0$. However, computationally determining $S_0$ is ineffective for many choices of $S$. We give an explicit and efficient algorithm to determine the set $S_0$ under certain Generalized Riemann Hypotheses, and implement the algorithm to determine $S_0$ when $S$ is the set of all odd integers.

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Ben Kane. 2009-03-17. Representing Sets with Sums of Triangular Numbers. https://doi.org/10.1093/imrn%2Frnp053

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