On finiteness theorems for sums of generalized polygonal numbers

2023/05/24 by Kane, Ben, Yang, Zichen
#11E25 #11F27 #11F30 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2305.14670

Abstract

In this paper, we consider mixed sums of generalized polygonal numbers. Specifically, we obtain a finiteness condition for universality of such sums; this means that it suffices to check representability of a finite subset of the positive integers in order to conclude that the sum of generalized polygonal numbers represents every positive integer. The sub-class of sums of generalized polygonal numbers which we consider is those sums of m_j-gonal numbers for which lcm(m₁-2,…,m_r-2)≤ \mathfrakM and we obtain a bound on the asymptotic growth of a constant Γ_\mathfrakM such that it suffices to check the representability condition for n≤ Γ_\mathfrakM.

Related

• Fermat's polygonal number theorem for repeated generalized polygonal numbers
• Almost universal ternary sums of polygonal numbers
• Weighted sums of generalized polygonal numbers with coefficients 1 or 2
• Conjectures of Sun about sums of polygonal numbers
• Sums of generalized polygonal numbers of almost prime "length"