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Erdos and Selfridge open their paper "The Product of Consecutive Integers is Never a Power" (1974) with the theorem

$\text{Theorem 1:}$ The product of two or more consecutive positive integers is never a power.

Question: While I am working through the paper to understand their argument, I wonder: has there been any work extending these results to other rings of integers? i.e. falsifying the equation in general $$ (a+1)\cdots(a+k)=b^l $$ for (what I suspect) some number field $F$, positive $a,b\in\mathcal{O}_F$, and $k,l\geq 2$.

Note: I know that for $F=\mathbb{Q}[\sqrt{5}],\>$ $a=b=\frac{1}{2}(\sqrt{5}-1),\>k=2,\>l=3$, the equation is true. Perhaps it would invite a reason to restrict the question to fields $F=\mathbb{Q}[\sqrt{m}]$ where $m\not\equiv 1\mod 4.$

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    $\begingroup$ There has been some study of exceptional units in number fields, that is, units $u$ such that $1+u$ is also a unit. This makes $u(1+u)$ a unit and a candidate for being a power of a unit. $\endgroup$ – Gerry Myerson Dec 5 '19 at 22:47
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It's not hard to find number fields where there are examples. E.g., the equation $(y-3)(y-2)(y-1)=y^2$, equivalently $y^3-7y^2+11y-6=0$, has a root $\alpha$, an algebraic integer, with $5<\alpha<6$, so if we let $x=\alpha-4$, then $x$ is a positive algebraic integer with $(x+1)(x+2)(x+3)=(x+4)^2$.

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  • $\begingroup$ Of course, there are even more trivial examples. Let $a$ be any positive algebraic integer, let $k,\ell$ be any integers at least two, let $b=((a+1)(a+2)\cdots(a+k))^{1/\ell}$, then you have an example in the ring of integers of the field generated by $a$ and $b$. $\endgroup$ – Gerry Myerson Dec 14 '19 at 23:02

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