Universal sums of generalized polygonal numbers of almost prime length

TL;DR

This paper proves finiteness theorems for universal sums of generalized polygonal numbers with inputs having a limited number of prime divisors, extending classical results to almost prime lengths.

Contribution

It establishes two finiteness theorems for universal sums of generalized polygonal numbers with inputs constrained by prime divisor counts, including an optimal bound for large prime divisor counts.

Findings

Finiteness theorems for universal sums with restricted prime divisors

Uniform bound independent of prime divisor count for large L

Optimal bound for finiteness check when L exceeds a multiple of log(m)

Abstract

In this paper, we consider universal sums of generalized polygonal numbers. Fixing $m\in\mathbb{N}_{\geq 3}$, we show two finiteness theorems for universal sums of generalized polygonal numbers whose inputs have a restricted number $L$ of prime divisors (counting multiplicity) away from an finite set of exceptional primes. In the first theorem, we fix $m$ and uniformly bound the finite check independent of $L\geq 900$, and in the second theorem, we give an optimal bound for the finiteness check if $L$ is larger than a constant times $\log(m)$.