Cubes from products of terms in progression with one term missing

TL;DR

This paper classifies all integer solutions to a specific product of linear terms (with one missing) being a perfect cube, using elliptic curve theory, and applies the results to rational points on a superelliptic curve.

Contribution

It provides a complete solution to a class of Diophantine equations involving products of linear terms with one missing, employing advanced elliptic curve techniques.

Findings

All solutions for the given product equation are determined.

The method applies elliptic curve Chabauty over number fields.

Answers a question about rational points on a superelliptic curve.

Abstract

Let $5 \leq k \leq 11$ and $0\leq i \leq k-1$ 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 $ny \neq 0$, $d\geq 1$, 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.