Computing the Residue Class of Partition Numbers

Abstract

In 1919, Ramanujan initiated the study of congruence properties of the integer partition function $p(n)$ by showing that $$p(5n+4) \equiv 0 \mod{5}$$ and $$p(7n+5) \equiv 0 \mod{7}$$ hold for all integers $n$. These results attracted a lot of interest in the mathematical community and inspired other mathematicians to investigate the divisibility of various classes of integer partitions.

The purpose of this thesis is to illustrate the use of generating series in the study of the residue classes of integer partition values. We begin by presenting the work of Mizuhara, Sellers and Swisher in 2015 on the residue classes of restricted plane partitions numbers. Next, we introduce Ramanujan's Conjecture regarding Ramanujan Congruences. Moreover, we use modular forms to present Ahlgren and Boylan's resolution of Ramanujan's Conjecture from 2003. Then, we discuss the open problems surrounding the distribution of the integer partitions values into residue classes and present Judge, Keith and Zanello's work from 2015 on the the distribution of the parity of the partition function. We continue by introducing $m-$ary partitions and provide an account of Andrews, Fraenkel and Sellers' work from 2015 and 2016 which yielded a complete characterization of the congruence classes of $m-$ary partitions with and without gaps. Finally, we present new results regarding the complete characterization of the residue classes of coloured $m-$ary partitions with and without gaps.