Powers from products of $k$ terms in progression: finiteness for small $k$

Michael A. Bennett¹

¹ Department of Mathematics University of British Columbia Vancouver, BC, Canada V6T 1Z2

Acta Arithmetica, Tome 184 (2018) no. 1, pp. 87-100.

Résumé

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.

Détail

DOI : 10.4064/aa171106-19-12

Keywords: striking result erd selfridge diophantine equation cdots k ell has solution positive integers ell min ell geq attempts derive analogous statement equation begin equation* label a cdots k ell tag * end equation* where similar nonexistence solutions has conjectured erd positive coprime suitably large have led large number interesting conditional results recently author jointly siksek proved fixed geq equation * has only finitely many solutions where neq ell geq variable gcd while here effectively computable explicitly determined certainly exceeds small values finiteness results * under coprimality assumptions have previously obtained leq author bruin hajdu the paper considerably extend using wide variety techniques prove theorem there exist finitely many integers ell gcd ell geq leq leq which equation * satisfied