A Computational Approach to Square Graham Partitions

Abstract

We propose an algorithm based on Monte Carlo Tree Search equipped with a supervised learned neural network to find an integer partition of n into squares satisfying n=lambda 1+lambda 2+& ctdot;+lambda l,1=1 lambda 1+1 lambda 2+& ctdot;+1 lambda l with squares lambda 1 >=lambda 2 >=& ctdot;>=lambda l >= 1 . By using induction and the search algorithm, we prove that such a partition exists if n >= 1508 .