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

exceptional unitsin 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