Cubes from products of terms in progression with one term missing

Abstract

Let $5\le k\le 11$ and $0\le i\le k-1$ be integers. We determine all solutions to the equation \begin{align*} n(n+d)(n+2d)\cdots(n+(i-1)d)(n+(i+1)d) \cdots (n+(k-1)d) = y^3 \end{align*} in integers $n,d,y$ with $ny\ne 0$, $d\ge 1$, and $\text{gcd}(n,d)=1$. Our method relies on the theory of elliptic curves, including elliptic curve Chabauty over a number field. As an application, we answer a question of Das, Laishram, Saradha, and Sharma concerning rational points on a certain superelliptic curve.

13 pages