| Abstrakt: |
In this paper, we present the results related to a problem posed by Andrzej Schinzel. Does the number $N_1(n)$ of integer solutions of the equation $$ \begin{align*}x_1+x_2+\cdots+x_n=x_1x_2\cdot\ldots\cdot x_n,\,\,x_1\ge x_2\ge\cdots\ge x_n\ge 1\end{align*} $$ tend to infinity with n ? Let a be a positive integer. We give a lower bound on the number of integer solutions, $N_a(n)$ , to the equation $$ \begin{align*}x_1+x_2+\cdots+x_n=ax_1x_2\cdot\ldots\cdot x_n,\,\, x_1\ge x_2\ge\cdots\ge x_n\ge 1.\end{align*} $$ We show that if $N_2(n)=1$ , then the number $2n-3$ is prime. The average behavior of $N_2(n)$ is studied. We prove that the set $\{n:N_2(n)\le k,\,n\ge 2\}$ has zero natural density. [ABSTRACT FROM AUTHOR] |
