The triangular theorem of eight and representation by quadratic polynomials

TL;DR

This paper explores the conditions under which sums of triangular numbers can represent all nonnegative integers, establishing a specific finite criterion and demonstrating the limitations of generalizing to quadratic polynomials with cross-terms.

Contribution

It proves a finite representability criterion for sums of triangular numbers and shows the non-existence of a similar finite criterion for quadratic polynomials with cross-terms.

Findings

Sum of triangular numbers represents all nonnegative integers iff it represents 1, 2, 4, 5, and 8.

No finite set of positive integers suffices for quadratic polynomials with cross-terms.

Finiteness theorems for positive definite quadratic forms do not extend to totally positive quadratic polynomials.

Abstract

We investigate here the representability of integers as sums of triangular numbers, where the $n$-th triangular number is given by $T_n = n(n + 1)/2$. In particular, we show that $f(x_1,x_2,..., x_k) = b_1 T_{x_1} +...+ b_k T_{x_k}$, for fixed positive integers $b_1, b_2,..., b_k$, represents every nonnegative integer if and only if it represents 1, 2, 4, 5, and 8. Moreover, if `cross-terms' are allowed in $f$, we show that no finite set of positive integers can play an analogous role, in turn showing that there is no overarching finiteness theorem which generalizes the statement from positive definite quadratic forms to totally positive quadratic polynomials.