On Cayley algorithm for double partition

TL;DR

This paper discusses a modification of Cayley's algorithm for double partition problems, providing a solution when the original conditions for the algorithm's success are not met.

Contribution

It introduces a modified algorithm that successfully reduces double partition problems even when Cayley's original conditions fail.

Findings

Modified algorithm solves reduction when original conditions are not satisfied

Algorithm effectively handles systems of two linear Diophantine equations

Enhances the applicability of Cayley's reduction method

Abstract

A double partition problem asks for a number of nonnegative integer solutions to a system of two linear Diophantine equations with integer coefficients. Artur Cayley suggested a reduction of a double partition to a sum of scalar partitions with an algorithm subject to a set of conditions. We show that when these conditions are not satisfied and the original algorithm fails its modification solves the reduction problem.