Almost universal mixed sums of squares and polygonal numbers

TL;DR

This paper characterizes when certain mixed sums of squares and polygonal numbers represent all but finitely many positive integers, using ternary quadratic forms for various parameters and primes.

Contribution

It provides a complete characterization of representability for mixed sums involving generalized polygonal numbers and squares, extending previous results.

Findings

Determines conditions for mixed sums to represent all but finitely many positive integers.

Uses ternary quadratic form theory to analyze representation problems.

Results depend on parameters and prime divisibility conditions.

Abstract

For each integer $m\ge3$, let $P_m(x)$ denote the generalized $m$-gonal number $\frac{(m-2)x^2-(m-4)x}{2}$ with $x\in\mathbb{Z}$. Given positive integers $a,b,c,k$ and an odd prime number $p$ with $p\nmid c$, we employ the theory of ternary quadratic forms to determine completely when the mixed sum $ax^2+by^2+cP_{p^k+2}(z)$ represents all but finitely many positive integers.