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

• 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. Dec 5, 2019 at 22:47

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 $$(x+1)(x+2)(x+3)=(x+4)^2$$.
• 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 $b$. Dec 14, 2019 at 23:02