Perfect powers in products of terms of elliptic divisibility sequences

TL;DR

This paper proves finiteness of solutions to a specific product equation involving elliptic divisibility sequences and provides an algorithm to find all solutions when the set of powers is known, advancing understanding in number theory.

Contribution

It establishes the finiteness of solutions for a class of Diophantine equations involving elliptic divisibility sequences and offers an explicit algorithm to determine all solutions.

Findings

Only finitely many solutions exist for the equation.

An algorithm can find all solutions given the set of powers.

The method is illustrated with a concrete example.

Abstract

Diophantine problems involving recurrence sequences have a long history and is an actively studied topic within number theory. In this paper, we connect to the field by considering the equation \begin{align*} B_mB_{m+d}\dots B_{m+(k-1)d}=y^\ell \end{align*} in positive integers $m,d,k,y$ with $\gcd(m,d)=1$ and $k\geq 2$, where $\ell\geq 2$ is a fixed integer and $B=(B_n)_{n=1}^\infty$ is an elliptic divisibility sequence, an important class of non-linear recurrences. We prove that the above equation admits only finitely many solutions. In fact, we present an algorithm to find all possible solutions, provided that the set of $\ell$-th powers in $B$ is given. (Note that this set is known to be finite.) We illustrate our method by an example.