Let
$$f(x, k, d) = x(x + d)\cdots(x + (k - 1)d)$$
be a polynomial with
$$k \geq 2, d \geq 1$$
. We consider the Diophantine
equation
$$\prod_{i=1}^{r} f(x_i, k_i, d) = y^{2}, r \geq 1$$
. Using the theory of Pell
equations, we affirm a conjecture of Bennett and van Luijk [3]; extend some results of this Diophantine equation for
$$d=1$$
, and give a positive answer to Question 3.2 of Zhang [19].
