Powers from products of $k$ terms in progression: finiteness for small $k$
Volume 184 / 2018
Abstract
A striking result of Erdős and Selfridge is that the Diophantine equation $$ n (n+1) \cdots (n + k-1) = y^{\ell} $$ has no solution in positive integers $n, k, y$ and $\ell$ with $\min \{ k, \ell \} \geq 2$. Attempts to derive an analogous statement for the equation \begin{equation*} \label{eq1-a} n (n+d) \cdots (n + (k-1)d) = y^{\ell},\tag{$*$} \end{equation*} where a similar nonexistence of solutions has been conjectured by Erdős to hold for $n$, $d$ positive and coprime and $k$ suitably large, have led to a large number of interesting conditional results.
Very recently, the author, jointly with Siksek, proved that, for fixed $k \geq k_0$, equation $(*)$ has only finitely many solutions (where $n, d, y \neq 0$ and $\ell \geq 2$ are variable, and $\gcd (n,d)=1$). While $k_0$ here is effectively computable, it is not explicitly determined and certainly exceeds $e^{10^6}$.
For small values of $k$, finiteness results for $(*)$ (under coprimality assumptions) have previously been obtained for $k \leq 82$ by the author, Bruin, Győry and Hajdu. The goal of the paper at hand is to considerably extend this by using a wide variety of new techniques. We prove
${\bf Theorem.}$ There exist at most finitely many integers $n, d, y, \ell$ and $k$ with $\gcd (n,d) = 1$, $\ell \geq 2$ and $4 \leq k \leq 15177$ for which equation $(*)$ is satisfied.
