On Geometric Progressions on Hyperelliptic Curves

Let C be a hyperelliptic curve over ${\mathbb Q}$ described by $y^2=a_0x^n+a_1x^{n-1}+\cdots+a_n$ , $a_i\in{\mathbb Q}$ . The points $P_{i}=(x_{i},y_{i})\in C(\mathbb{Q} )$ , $i=1,2,\ldots,k$ , are said to be in a geometric progression of length k if the rational numbers x_i, $i=1,2,\ldots,k$ , form a geometric progression sequence in ${\mathbb Q}$ , i.e., x_i = ptⁱ for some $p,t\in{\mathbb Q}$ . In this paper we prove the existence of an infinite family of hyperelliptic curves on which there is a sequence of rational points in a geometric progression of length at least eight.

Received February 18 2016; revised versions received June 14 2016; June 17 2016. Published in Journal of Integer Sequences, June 29 2016.

On Geometric Progressions on Hyperelliptic Curves

Mohamed Alaa Department of Mathematics Faculty of Science Cairo University Giza Egypt Mohammad Sadek Mathematics and Actuarial Science Department American University in Cairo AUC Avenue New Cairo Egypt

Mohamed Alaa
Department of Mathematics
Faculty of Science
Cairo University
Giza
Egypt
Mohammad Sadek
Mathematics and Actuarial Science Department
American University in Cairo
AUC Avenue
New Cairo
Egypt