On finiteness theorems for sums of generalized polygonal numbers

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 i {m_{i}} -gonal numbers for which lcm ⁢ ( m 1 - 2 , … , m r - 2 ) ≤ 𝔐 {\text{lcm}(m_{1}-2,\dots,m_{r}-2)\leq\mathfrak{M}} and we obtain a bound on the asymptotic growth of a constant Γ 𝔐 {\Gamma_{\mathfrak{M}}} such that it suffices to check the representability condition for n ≤ Γ 𝔐 {n\leq\Gamma_{\mathfrak{M}}} .