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:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>m</m:mi> <m:mi>i</m:mi> </m:msub> </m:math> {m_{i}} -gonal numbers for which <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mtext>lcm</m:mtext> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msub> <m:mi>m</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>-</m:mo> <m:mn>2</m:mn> </m:mrow> <m:mo>,</m:mo> <m:mi mathvariant="normal">…</m:mi> <m:mo>,</m:mo> <m:mrow> <m:msub> <m:mi>m</m:mi> <m:mi>r</m:mi> </m:msub> <m:mo>-</m:mo> <m:mn>2</m:mn> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>≤</m:mo> <m:mi>𝔐</m:mi> </m:mrow> </m:math> {\text{lcm}(m_{1}-2,\dots,m_{r}-2)\leq\mathfrak{M}} and we obtain a bound on the asymptotic growth of a constant <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="normal">Γ</m:mi> <m:mi>𝔐</m:mi> </m:msub> </m:math> {\Gamma_{\mathfrak{M}}} such that it suffices to check the representability condition for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>≤</m:mo> <m:msub> <m:mi mathvariant="normal">Γ</m:mi> <m:mi>𝔐</m:mi> </m:msub> </m:mrow> </m:math> {n\leq\Gamma_{\mathfrak{M}}} .