Universal sums of polygonal numbers and generalized polygonal numbers

Abstract

In this thesis, we investigate the representations of integers as sums of polygonal and generalized polygonal numbers. Specifically, the thesis focuses on the sums of generalized heptagonal numbers and that of hexagonal numbers, with positive integer coefficients. Such sums are classified to be universal if they represent every positive integer. The fifteen theorem of Conway-Schneeberger states that a sum of squares being universal is equivalent to that sum representing all positive integers up to 15. Similar generalizations by Bosma and Kane showed that sums of triangular numbers (or generalized hexagonal numbers) is universal if and only if it represents all positive integers up to 8. Ju proved a similar result for generalized pentagonal number sums and Ju and Oh proved a similar result for generalized octagonal number sums. In this thesis, we aim to prove and find an explicit finite upper bound such that for any given sum of generalized heptagonal numbers, it is universal if it represents all positive integers up the given finite bound. We employ analytical methods to compute such an explicit upper bound, which however is not the smallest such bound. We also extend the methods used to show further computations and developments towards sums of hexagonal numbers.