# Extension of Erdos-Selfridge Theorem

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 $$m\not\equiv 1\mod 4.$$