Abstract
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 \({\prod_{i = 1}^{r} f(x_i, k_i, d) = y^2}\), which is inspired by a question of Erdős and Graham [4, p. 67]. Using the theory of Pellian equation, we give infinitely many (nontrivial) positive integer solutions of the above Diophantine equation for some cases.
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This research was supported by the National Natural Science Foundation of China (Grant No. 11501052).
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Zhang, Y. On products of consecutive arithmetic progressions. II. Acta Math. Hungar. 156, 240–254 (2018). https://doi.org/10.1007/s10474-018-0850-7
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DOI: https://doi.org/10.1007/s10474-018-0850-7