On Diophantine Equations of the Form $(x-a_1)(x-a_2)ldots(x-a_k)+r=y^n$

摘要

Erd?‘s and Selfridge [3] proved that a product of consecutive integers can never be a perfect power. That is, the equation $x(x+1)(x+2)ldots(x+(m-1))=y^n$ has no solutions in positive integers $x,m,n$ where $m,n1$ and $yin Q$. We consider the equation$$(x-a_1)(x-a_2)ldots(x-a_k)+r=y^n$$where $0a‰¤ a_1 a_2 cdots a_k$ are integers and, with $rin Q,na‰￥ 3$ and we prove a finiteness theorem for the number of solutions e?‘￥ in $Z,y$ in e?‘?. Following that, we show that, more interestingly, for every nonzero integer e?‘?2 and for any nonzero integer e?‘? which is not a perfect e?‘?-th power for which the equation admits solutions, e?‘? is bounded by an effective bound.