Erdos and Selfridge open their paper ["The Product of Consecutive Integers is Never a Power"](https://projecteuclid.org/download/pdf_1/euclid.ijm/1256050816) (1974) with the theorem



> 
> $\text{Theorem 1:}$ *The product of two or more consecutive positive integers is never a power.*
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**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 $m\not\equiv 1\mod 4.$



Extension of Erdos-Selfridge Theorem
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 $(x+1)(x+2)(x+3)=(x+4)^2$. 


