Tight universal sums of $m$-gonal numbers

TL;DR

This paper establishes the existence of minimal criteria for sums of generalized m-gonal numbers to be tight universal over all integers greater than or equal to n, introducing an algorithm for classification and providing experimental results.

Contribution

It proves the existence of minimal tight universality criteria for sums of generalized m-gonal numbers and introduces an algorithm for their classification.

Findings

Existence of minimal tight universality criterion sets for all (m,n) pairs.

An algorithm to identify all candidate sums for given (m,n).

Experimental classification results of tight universal sums.

Abstract

For a positive integer $n$, the set of all integers greater than or equal to $n$ is denoted by $\mathcal T(n)$. A sum of generalized $m$-gonal numbers $g$ is called tight $\mathcal T(n)$-universal if the set of all nonzero integers represented by $g$ is equal to $\mathcal T(n)$. In this article, we prove the existence of a minimal tight $\mathcal T(n)$-universality criterion set for a sum of generalized $m$-gonal numbers for any pair $(m,n)$. To achieve this, we introduce an algorithm giving all candidates for tight $\mathcal T(n)$-universal sums of generalized $m$-gonal numbers for any given pair $(m,n)$. Furthermore, we provide some experimental results on the classification of tight $\mathcal T(n)$-universal sums of generalized $m$-gonal numbers.