Golden Ratio Based Partitions of the Integers

Abstract

A partition of the integers is a collection of pairwise disjoint integer subsets whose union contains every integer. The upper and lower Wythoff sequences form one such partition using the golden ratio. This construction is notable among many things for providing the winning positions in the two-heap game of Nim. This work provides a generalization of the Wythoff partition involving an arbitrary number of sets and analyzes various properties of specific cases of this generalization.