Cubes from products of terms in progression with one term missing
Kyle Pratt
TL;DR
The paper determines all integer solutions to $\prod_{j\neq i}^{k-1}(n+jd)=y^3$ for $5\le k\le 11$ and $0\le i\le k-1$ under $\gcd(n,d)=1$, $d\ge1$, and $ny\neq0$. It develops a framework based on factoring $n+jd=a_jx_j^3$ with cube-free $a_j$ (coefficient vectors), reduces the problem to finite families of ternary cubic equations, and eliminates most cases via Selmer-type results and elliptic curves, aided by heavy computational verification. The remaining critical instance $k=7,i=3$ is resolved by translating a pair of cubics into an elliptic curve over the cubic field $K=\mathbb{Q}(\sqrt[3]{5})$ and applying the elliptic-curve Chabauty method, yielding two new solutions and thus completing the classification. As a corollary, the paper determines all rational points on a related superelliptic curve, illustrating the method’s power in studying rational points on higher-degree curves. Overall, the work demonstrates how combining factorization structure, ternary cubic analysis, and Chabauty over number fields can solve intricate Diophantine problems with arithmetic-progression products.
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.