Constant-Sum Partitions of Even Cardinality

Abstract

A number n is said to have the constant-sum property if for every possible partition of n, there exists disjoint subsets of the first n integers, such that the sum of the elements in each subset have the same remainder when divided by n (This means they are congruent with some constant integer c modulo n).

For the case when n is odd, Kaplan, Lev, and Roditty [1] proved in 2009 that there exists a constant-sum partition for n if and only if only there exists no more than one singleton set. For the case when n is even and split into an odd number of subsets, Freyberg [2] proved in 2019 that an n/2-sum partition exists for n if and only if there exists no more than one singleton subset.

This leaves open the case when n is even and each subset has an even number of elements. My UROP project focused on the case when n is even and broken into groups that each have even cardinality, that is, they each have an even number of elements.