This is a follow-up to this question (and comments thereon). Namely, it follows from Felipe Voloch's comment that for any $n>2$ there is a finite set of integral $(x, y),$ such that $$ \prod_{i=1}^n(x+i) = n y^2.$$ The question is: is there any $n$ for which the explicit list is known (and if so, how it was generated)? I would assume that $n=3, 4$ should be tractable...

**ADDENDUM** In joro's answer he points out that for $n=3$ there are no nontrivial solutions. For $n=4,$ sage computes the Weierstrass form of the curve as
$$y^2 = x^3 - 208/3*x + 4480/27.$$
It then finds that the rank of the MW group is $0,$ and that the torsion subgroup is of the form $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/4\mathbb{Z}.$ The generators are claimed to be $(-4/3: 16:1)$ and $(8/3:0:1).$ I assume this is for the transformed $x, y,$ so I am not sure what this means for the integral solutions - I am sure someone here does, though...

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