Universal Sums of Triangular Numbers and Squares

TL;DR

This paper characterizes all universal sums of triangular numbers and squares, identifying the specific numbers they must represent to be considered universal, thus advancing the understanding of figurate number representations.

Contribution

It provides a complete characterization of universal sums involving triangular numbers and squares, specifying the exact numbers they must represent.

Findings

Identifies the specific numbers that universal sums must represent.

Proves the equivalence between universality and representing a finite set of numbers.

Advances the theory of figurate number representations.

Abstract

In this paper, we study universal sums of triangular numbers and squares. Specifically, we prove that a sum of triangular numbers and squares is universal if and only if it represents $1,2,3,4,5,6,7,8,10,13,14,15,18,19,20,23,27,28,34,41,47$, and $48$.