An Algorithm to Solve the Equal-Sum-Product Problem

TL;DR

This paper introduces a recursive algorithm for solving a class of Diophantine equations related to finding positive integer n-tuples with equal sum and product, and explores its implementation and number-theoretic connections.

Contribution

It presents a novel recursive algorithm for the equal-sum-product problem and discusses its implementation using Binary Search Trees, also linking solutions to Sophie Germain primes.

Findings

The algorithm successfully finds all solutions to the problem.

Implementation with Binary Search Trees enhances computational efficiency.

Connections between solutions and Sophie Germain primes are established.

Abstract

A recursive algorithm is constructed which finds all solutions to a class of Diophantine equations connected to the problem of determining ordered n-tuples of positive integers satisfying the property that their sum is equal to their product. An examination of the use of Binary Search Trees in implementing the algorithm into a working program is given. In addition an application of the algorithm for searching possible extra exceptional values of the equal-sum-product problem is explored after demonstrating a link between these numbers and the Sophie Germain primes.