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arXiv · 2105.02792

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

Pranabesh Das·Shanta Laishram·N. Saradha·Divyum Sharma
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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 $4 \leq k \leq 8.$

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Pranabesh Das, Shanta Laishram, N. Saradha, Divyum Sharma. 2021-05-06. Rational solutions to the Variants of Erd\H{o}s- Selfridge superelliptic curves. https://arxiv.org/abs/2105.02792

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