An algorithm for the Product-Sum Equality

TL;DR

This paper introduces a recursive algorithm to find all finite sequences of positive integers where the product equals the sum, using an efficient iterative extension method.

Contribution

The paper presents a novel recursive algorithm that efficiently identifies all such sequences, improving upon previous methods in terms of computational complexity.

Findings

Algorithm successfully finds all sequences with product equal to sum

Time complexity comparable to quicksort, indicating high efficiency

Applicable to problems involving integer sequences with specific product-sum constraints

Abstract

We propose a recursive algorithm for identifying all finite sequences of positive integers whose product equals their sum. Our method uses solutions of strictly shorter length that are iteratively extended in pursuit of a valid solution. The algorithm is efficient, with a time complexity similar to the quick sort algorithm.