A class of Diophantine equations involving Bernoulli polynomials
Let \(a,b\in\mathbb{Q}\backslash\{0\}\) and \(C(y)\in\mathbb{Q}[(y]\). The authors consider the Diophantine equations \[ aB_m(x)=bf_n(y)+C(y) \] and \[ af_m(x)=bB_n(y)+C(y) \] with \(m\geq n>\text{deg}\;C+2\) for solutions in integers \(x,y\). Here \(f_n(y)=x(x+1)\ldots (x+n-1)\) and \(B_n(x)\) denote the Bernoulli polynomials. The main theorems give finiteness results by applying the method of \textit{Y. F. Bilu} and \textit{R. F. Tichy} [Acta Arith. 95, No.~3, 261--288 (2000; Zbl 0958.11049)].
scientific article; zbMATH DE number 2175008