Equal-Sum-Product problem II

Zakarczemny, Maciej 

Canadian mathematical bulletin, Tome 67 (2024) no. 3, pp. 582-592

Résumé

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.

Détail

DOI : 10.4153/S000843952300098X

Mots-clés : Equal-sum-product, exceptional set, natural density