Geodesic
On the equal sum and product problem
Maciej Szymon Zakarczemny1 ; Maciej Szymon Zakarczemny
1Department of Applied Mathematics, Faculty of Computer Science and Telecommunications, Cracow University of Technology, Cracow, Poland
Acta mathematica Universitatis Comenianae, Tome 90 (2021) no. 4, pp. 387-402 
Maciej Szymon Zakarczemny; Maciej Szymon Zakarczemny. On the equal sum and product problem. Acta mathematica Universitatis Comenianae, Tome 90 (2021) no. 4, pp. 387-402. http://geodesic.mathdoc.fr/item/AMUC_2021_90_4_a2/
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     title = { On the equal sum and product problem},
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     pages = {387--402},
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The paper presents the results which are connected with the following problem formulated by Andrzej Schinzel.Does the number $N(n)$ of integer solutions of the equation $x_1+x_2+\dots+x_n=x_1x_2\cdots x_n$ satisfying $1\le x_1\le x_2\le\dots\le x_n$ tend to infinity with $n$? We give a general lower bound on $N(n)$. We obtain an $\mathrm{\Omega}$-estimate for $\frac{1}{x}\sum_{1. We provide necessary conditions for $n$ to be in the exceptional set $\{n:N(n)=1,\,n\ge 2\}$. Using elementary methods, we show that if $N(n)=2$, then $n-1, 2n-1\in\{p,p^2,p^3,pq\}$, where $p,q$ are prime numbers. We prove that the set $\{n:N(n)\le k, n\ge 2\}$, and the exceptional set have zero natural density. We give new bounds on sum of coordinates of not-typical solutions. We prove that the system of equations of the equal-sum-product problem has a finite number of solutions.