Representing Sets with Sums of Triangular Numbers

TL;DR

This paper studies sums of triangular numbers representing sets of integers, providing an explicit algorithm under certain hypotheses to determine minimal representing subsets, with implementation for odd integers.

Contribution

It introduces an efficient algorithm, under Generalized Riemann Hypotheses, to find minimal subsets representing sets via sums of triangular numbers.

Findings

Algorithm successfully determines the minimal set for odd integers

Provides explicit criteria under Galois hypotheses for representation

Demonstrates the method's effectiveness through implementation

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.