Almost universal ternary sums of pentagonal numbers

TL;DR

This paper investigates conditions under which certain weighted sums of generalized pentagonal numbers can represent all but finitely many positive integers, using the theory of ternary quadratic forms.

Contribution

It provides a characterization of when these sums are almost universal, extending the understanding of representations involving pentagonal numbers and powers of two.

Findings

Identifies specific conditions for almost universality.

Classifies sums that represent all but finitely many positive integers.

Uses advanced quadratic form theory to derive results.

Abstract

For each integer $x$, the $x$-th generalized pentagonal number is denoted by $P_5(x)=(3x^2-x)/2$. Given odd positive integers $a,b,c$ and non-negative integers $r,s$, we employ the theory of ternary quadratic forms to determine when the sum $aP_5(x)+2^rbP_5(y)+2^scP_5(z)$ represents all but finitely many positive integers.