Problem #363: Is it true that there are only finitely many collections of...

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Progress

Erdős and Selfridge have proved that the product of consecutive integers is never a power. The condition $\lvert I_i\rvert \geq 4$ is necessary here, since Pomerance has observed that the product of\[(2^{n-1}-1)2^{n-1}(2^{n-1}+1),\]\[(2^n-1)2^n(2^n+1),\]\[(2^{2n-1}-2)(2^{2n-1}-1)2^{2n-1},\]and\[(2^{2n}-2)(2^{2n}-1)2^{2n}\]is always a square.

This is false: Ulas [Ul05] has proved there are infinitely many solutions when $n=4$ or $n\geq 6$ and $\lvert I_i\rvert=4$ for $1\leq i\leq n$. Bauer and Bennett [BaBe07] proved there are infinitely many solutions when $n=3$ or $n=5$ and $\lvert I_i\rvert=4$ for $1\leq i\leq n$. Furthermore, Bennett and Van Luijk [BeVL12] have found infinitely many solutions when $n\geq 5$ and $\lvert I_i\rvert=5$ for $1\leq i\leq n$.

In general, Ulas conjectures there are infinitely many solutions for any fixed size of $\lvert I_i\rvert$, provided $n$ is sufficiently large.

See also [930] for a more general question.