Cubes from products of terms in progression with one term missing

2025/12/19 by Kyle Pratt, Pratt, Kyle
Mathematics · Computer Science · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Cryptography and Residue Arithmetic

paper · doi:10.48550/arxiv.2512.17821

Abstract

Let 5 ≤ k ≤ 11 and 0≤ i ≤ k-1 be integers. We determine all solutions to the equation n(n+d)(n+2d)⋯(n+(i-1)d)(n+(i+1)d) ⋯ (n+(k-1)d) = y³ in integers n,d,y with ny ≠ 0, d≥ 1, and 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.

Citations

• A conjecture of Erdos, supersingular primes and short character sums

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