Rational solutions to the variants of Erdős–Selfridge superelliptic curves

Abstract

For the superelliptic curves of the form [Formula: see text] with [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] a prime, Das, Laishram, Saradha and Edis showed that the superelliptic curve has no rational points for [Formula: see text]. In fact, the double exponential bound, obtained in these papers is far from the reality. In this paper, we study the superelliptic curves for small values of [Formula: see text]. In particular, we explicitly solve the above equation for [Formula: see text]