## More Congruences for the k-regular Partition Function

##### Abstract

A partition of a number n is a list of positive integers whose sum is n. For example,
4 + 2 + 1 and 4 + 1 + 1 + 1 are both partitions of 7. It can be shown that 4 has 5
partitions, 9 has 30 partitions, 14 has 135 partitions, and Srinivasa Ramanujan proved
the following beautiful result: the number of partitions of 5n + 4 is divisible by 5
for any nonnegative integer n. The k-regular partition function counts the number of
partitions of n whose parts are not divisible by k. In 2012, for particular values of k,
David Furcy and David Penniston found many families of integers whose number of
k-regular partitions is divisible by 3. In this paper, I extend their results to larger values
of k and provide an overview of the methodology used to arrive at the result. In the
interest of brevity, only a sketch of the proof is given.

##### Subject

Number theory

Mathematics

Arithmetic

Geometry