Glasnik Matematicki, Vol. 52, No. 1 (2017), 1-10.

GEOMETRIC PROGRESSIONS ON ELLIPTIC CURVES

Abdoul Aziz Ciss and Dustin Moody

Laboratoire de Traitement de l'Information et Systèmes Intelligents,, École Polytechnique de Thiès, BP A10 Thiès, Sénégal
e-mail: aaciss@ept.sn

National Institute of Standards and Technology (NIST), 100 Bureau Drive, Gaithersburg, 20899-8930, USA
e-mail: dustin.moody@nist.gov


Abstract.   In this paper, we look at long geometric progressions on different models of elliptic curves, namely Weierstrass curves, Edwards and twisted Edwards curves, Huff curves and general quartics curves. By a geometric progression on an elliptic curve, we mean the existence of rational points on the curve whose x-coordinates (or y-coordinates) are in geometric progression. We find infinite families of twisted Edwards curves and Huff curves with geometric progressions of length 5, an infinite family of Weierstrass curves with 8-term progressions, as well as infinite families of quartic curves containing 10-term geometric progressions.

2010 Mathematics Subject Classification.   11B25, 11D41, 11G05.

Key words and phrases.   Arithmetic progression, geometric progression, elliptic curves.


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DOI: 10.3336/gm.52.1.01


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