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].

中文翻译：

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关于连续算术级数的乘积。三级

令\（f（x，k，d）= x（x + d）\ cdots（x +（k-1）d）\）是\（k \ geq 2，d \ geq 1 \）的多项式 。我们考虑Diophantine方程\（\ prod_ {i = 1} ^ {r} f（x_i，k_i，d）= y ^ {2}，r \ geq 1 \）。使用Pell方程的理论，我们肯定Bennett和van Luijk [3]的猜想。扩展了Diophantine方程\（d = 1 \）的一些结果，并对张[19]的问题3.2给出了肯定的答案。