On Diophantine Equations of the Form $(x-a_1)(x-a_2)\ldots(x-a_k)+r=y^n$
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Erdős 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 $m,n>1$ and $y\in Q$. We consider the equation
$$(x-a_1)(x-a_2)\ldots(x-a_k)+r=y^n$$
where $0\leq a_1 < a_2 <\cdots < a_k$ are integers and, with $r\in Q,n\geq 3$ and we prove a finiteness theorem for the number of solutions 𝑥 in $Z,y$ in 𝑄. Following that, we show that, more interestingly, for every nonzero integer $n>2$ and for any nonzero integer 𝑟 which is not a perfect 𝑛-th power for which the equation admits solutions, 𝑘 is bounded by an effective bound.
Manisha Kulkarni^{1} ^{} B Sury^{2} ^{}
Volume 133, 2023
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