Representing Sets with Sums of Triangular Numbers

Details Representing Sets with Sums of Triangular Numbers Representing Sets with Sums of Triangular Numbers

2009-03-17

arxiv.org/abs/0903.3026v1

Int. Math. Res. Not., Vol. 2009, No. 17, pp. 3264-3285

Mathematics

Number Theory

18 pages, 3 tables

Scientific paper

10.1093/imrn/rnp053

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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