Cubes from products of terms in progression with one term missing

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.

💡 Research Summary

The paper studies the Diophantine equation obtained by taking the product of all terms of an arithmetic progression of length k, omitting a single term, and demanding that the result be a perfect cube. Formally, for integers 5 ≤ k ≤ 11 and 0 ≤ i ≤ k‑1, the authors seek all integer triples (n, d, y) with ny ≠ 0, d ≥ 1, gcd(n,d)=1 satisfying
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