A Diophantine transport problem from 2016 and its possible solution in 1903

TL;DR

This paper reviews classical methods for solving systems of linear Diophantine equations and inequalities, highlighting Elliott's 1903 method and its development, and applies these to a 2016 transport problem using formal power series.

Contribution

It revisits and applies Elliott's 1903 method within the context of a modern transport problem, illustrating the use of partition analysis and formal power series.

Findings

Solution expressed as a formal power series in several variables.

Expansion of a rational function of a special form.

Demonstrates the applicability of classical Diophantine methods to modern problems.

Abstract

Motivated by a recent Diophantine transport problem about how to transport profitably a group of persons or objects, we survey classical facts about solving systems of linear Diophantine equations and inequalities in nonnegative integers. We emphasize on the method of Elliott from 1903 and its further developed by MacMahon in his ``$\Omega$-Calculus'' or Partition Analysis. As an illustration we obtain the solution of the considered transport problem in terms of a formal power series in several variables which is an expansion of a rational function of a special form.