On the equation \(f(x+1)\dots f(x+k)= f(y+1)\dots f(y+mk)\)
The authors study the integral solutions of the equation \[ f(x+1)\cdot\dots\cdot f(x+k)= f(y+1)\cdot \dots\cdot f(y+m k) \tag{*} \] when \(f\) is a power of a monic irreducible polynomial with rational coefficients and \(k\geq 2\), \(m\geq 2\) are integral parameters. A special case is \(f(x)=x\), previously studied by \textit{N. Saradha} and the second named author [Indag. Math., New Ser. 3, 79-90 (1992; Zbl 0757.11011)]. In that paper it was proved that, for integers \(x\geq 0\), \(y\geq 0\), \(k\geq 2\) and \(m\geq 2\), the equation \((x+1)\dots (x+k)= (y+1)\dots (y+mk)\) implies that \(\max(x,y,k)\) is bounded by an effective constant depending only on \(m\). In the present paper results of an analogous type are proved for equation (*), provided that certain reasonable assumptions are fulfilled. Remark. In the reviewer's opinion, it is worthwhile in papers like the one under review to -- at least -- discuss the possibility of making the stated results explicit.
scientific article; zbMATH DE number 510025