Rational solutions to the variants of Erdos-Selfridge superelliptic curves

Article Type

Research Article

Publication Title

International Journal of Number Theory

Abstract

For the superelliptic curves of the form (x + 1)⋯(x + i - 1)(x + i + 1)⋯(x + k) = yâ.,"with x,y â., , y≠0, k ≥ 3, 1 ≤ i ≤ k, â.,"≥ 2, a prime, Das, Laishram, Saradha and Edis showed that the superelliptic curve has no rational points for â.,"≥ e3k. 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 k. In particular, we explicitly solve the above equation for 4 ≤ k ≤ 8.

First Page

1707

Last Page

1744

DOI

https://10.1142/S1793042123500835

Publication Date

8-1-2023

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