Equal Sum and Product Problem III

TL;DR

This paper investigates the solutions to the equal sum and product problem, providing an asymptotic formula for their count and demonstrating that the solution count can be significantly larger than its average, highlighting interesting growth behavior.

Contribution

The paper introduces an asymptotic formula for the sum of solutions up to a certain size and shows that the solution count can be much larger than average, revealing new insights into the problem's growth.

Findings

Asymptotic formula for  sum of solutions for  range of n.

Solution counts are significantly larger than their average, indicating high variability and growth in the number of solutions.

Abstract

Denote by $N(n)$ the number of integer solutions $(x_1,\,x_2,\ldots ,x_n)$ of the equation $x_1+x_2+\ldots+x_n=x_1x_2\cdot\ldots\cdot x_n$ such that $x_1\ge x_2\ge\ldots\ge x_n\ge 1$, $n \in \mathbb{Z}^+$. The aim of this paper are is twofold: first we present an asymptotic formula for $\sum\limits_{2\le n\le x}N(n)$, then we verify that the counting function $N(n)$ takes very large value compared to its average value.