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

• # Fulltext

https://www.ias.ac.in/article/fulltext/pmsc/121/03/0245-0247

• # Keywords

Diophantine equations; Erdős–Selfridge.

• # Abstract

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 $m,n&gt;1$ and $y\in Q$. We consider the equation

$$(x-a_1)(x-a_2)\ldots(x-a_k)+r=y^n$$

where $0\leq a_1 &lt; a_2 &lt;\cdots &lt; a_k$ 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 $n&gt;2$ 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.

• # Author Affiliations

1. Poornaprajna Institute of Scientific Research, Devanahalli, Bangalore 562 110, India
2. Statistics & Mathematics Unit, Indian Statistical Institute, 8th Mile, Mysore Road, Bangalore 560 059, India

• # Proceedings – Mathematical Sciences

Current Issue
Volume 129 | Issue 5
November 2019

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019