Completing an universal $m$-gonal form with a closing unary piece

TL;DR

This paper demonstrates that for any $m$-gonal form with $m \,\geq\, 12$ representing all positive integers up to $m-4$, adding a unary $m$-gonal form can produce a universal form that represents all positive integers.

Contribution

It introduces a method to complete an $m$-gonal form to a universal form by adding a unary $m$-gonal form, for all $m \,\geq\, 12$.

Findings

Any $m$-gonal form with $m \,\geq\, 12$ representing integers up to $m-4$ can be completed to a universal form.

Adding a unary $m$-gonal form suffices to achieve universality.

The approach generalizes the construction of universal $m$-gonal forms.

Abstract

In this paper, we show that for any $m$-gonal form $F_m(\mathbf x)$ with $m \ge 12$ which represents every positive integer up to $m-4$, by putting together only unary $m$-gonal form, we may complete an universal form.