Rational solutions to the Variants 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 $x,y\in \mathbb{Q}$, $y\ne 0$, $k\ge 3$, $1\le i\le k$, $\ell \ge 2,$ a prime, Das, Laishram, Saradha, and Edis showed that the superelliptic curve has no rational points for $\ell \ge e^{3^k}$. 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 $4\le k\le 8.$

arXiv comment39 pages, To appear in the International Journal of Number Theory