Conjectures of Sun About Sums of Polygonal Numbers

In this paper, we consider representations of positive integers as sums of generalized m-gonal numbers, which extend the formula for the number of dots needed to make up a regular m-gon. We mainly restrict to the case where the sums contain at most four distinct generalized m-gonal numbers, with the second m-gonal number repeated twice, the third repeated four times, and the last is repeated eight times. For a number of small choices of m, Sun conjectured that every positive integer may be written in this form. By obtaining explicit quantitative bounds for Fourier coefficients related to theta functions which encode the number of such representations, we verify that Sun's conjecture is true for sufficiently large positive integers. Since there are only finitely many choices of m appearing in Sun's conjecture, this reduces Sun's conjecture to a verification of finitely many cases. Moreover, the bound beyond which we prove that Sun's conjecture holds is explicit.