Geodesic
On geometric progressions on hyperelliptic curves
Journal of integer sequences, Tome 19 (2016) no. 6.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: 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., $x_{i} = pt^{i}$ 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.
Classification : 14G05, 11B83
Keywords: geometric progression, hyperelliptic curve, rational point 
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     author = {Alaa, Mohamed and Sadek, Mohammad},
     title = {On geometric progressions on hyperelliptic curves},
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     year = {2016},
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Alaa, Mohamed; Sadek, Mohammad. On geometric progressions on hyperelliptic curves. Journal of integer sequences, Tome 19 (2016) no. 6. http://geodesic.mathdoc.fr/item/JIS_2016__19_6_a4/