On the sums of three generalized polygonal numbers

Abstract

For each natural number $m\ge 3$, let $P_m(x)$ denote the generalized $m$-gonal number $\frac{(m-2)x^2-(m-4)x}{2}$ with $x\in\mathbb{Z}$. In this paper, with the help of the congruence theta function, we establish conditions on $a$, $b$, $c$ for which the sum $P_a(x)+P_b(y)+P_c(z)$ represents all but finitely many positive integers.