Rational solutions to the Variants of Erd\H{o}s- Selfridge superelliptic curves

TL;DR

This paper investigates rational solutions to certain superelliptic curves related to Erdős-Selfridge variants, providing explicit solutions for small values of k (4 to 8) and improving understanding beyond previous double exponential bounds.

Contribution

It explicitly solves the superelliptic equations for small k (4 to 8), advancing beyond prior bounds and offering concrete solutions for these cases.

Findings

Explicit solutions for k=4 to 8

Improved bounds on rational points for small k

Enhanced understanding of Erdős-Selfridge superelliptic curves

Abstract

For the superelliptic curves of the form $$ (x+1) \cdots(x+i-1)(x+i+1)\cdots (x+k)=y^\ell$$ with $x,y \in \mathbb{Q}$, $y\neq 0$, $k \geq 3$, $1\leq i\leq k$, $\ell \geq 2,$ a prime, Das, Laishram, Saradha, and Edis showed that the superelliptic curve has no rational points for $\ell\geq e^{3^k}$. In fact, the double exponential bound, obtained in these papers is far from reality. In this paper, we study the superelliptic curves for small values of $k$. In particular, we explicitly solve the above equation for $4 \leq k \leq 8.$