On the Solutions of Three Variable Frobenius Related Problems Using Order Reduction Approach

TL;DR

This paper introduces order reduction methods to efficiently determine the number and solutions of three-variable Frobenius related problems, simplifying complex equations into more manageable forms.

Contribution

It proposes two novel order reduction techniques for solving three-variable Frobenius problems, including an algorithm and a conjecture on solutions.

Findings

Reduction from three-variable to two-variable systems

Development of an algorithm for order two equations

Formulation of an open problem and conjecture

Abstract

This paper presents a new approach to determine the number of solutions of three variable Frobenius related problems and to find their solutions by using order reducing methods. Here, the order of a Frobenius related problem means the number of variables appearing in the problem. We present two types of order reduction methods that can be applied to the problem of finding all nonnegative solutions of three variable Frobenius related problems. The first method is used to reduce the equation of order three from a three variable Frobenius related problem to be a system of equations with two fixed variables. The second method reduces the equation of order three into three equations of order two, for which an algorithm is designed with an interesting open problem on solutions left as a conjecture.