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GEOMETRIC PROGRESSIONS ON ELLIPTIC CURVES.

Abdoul Aziz Ciss1, Dustin Moody2

  • 1Laboratoire de Traitement de l'Information et Systèmes Intelligents, Ècole Polytechnique de Thiès, BP A10 Thiès, Sénégal, aaciss@ept.sn.

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Summary

Researchers explored long geometric progressions on various elliptic curves. They discovered infinite families of curves, including twisted Edwards and Huff curves, exhibiting progressions up to 10 terms.

Keywords:
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Area of Science:

  • Number Theory
  • Algebraic Geometry
  • Elliptic Curves

Background:

  • Elliptic curves are fundamental objects in number theory with applications in cryptography.
  • Geometric progressions involve sequences where each term after the first is found by multiplying the previous one by a fixed, non-zero number.
  • Investigating rational points on elliptic curves is a key area of research.

Purpose of the Study:

  • To investigate the existence of long geometric progressions of rational points on different models of elliptic curves.
  • To identify specific types of elliptic curves that admit such progressions.
  • To determine the maximum length of geometric progressions found on these curves.

Main Methods:

  • Analysis of rational points on Weierstrass, Edwards, twisted Edwards, Huff, and general quartic curves.
  • Definition of geometric progression based on the x- or y-coordinates of rational points.
  • Construction of infinite families of curves exhibiting geometric progressions.

Main Results:

  • Identified infinite families of twisted Edwards and Huff curves with geometric progressions of length 5.
  • Found an infinite family of Weierstrass curves admitting 8-term geometric progressions.
  • Discovered infinite families of quartic curves containing 10-term geometric progressions.

Conclusions:

  • The study demonstrates the existence of significant geometric progressions on various elliptic curve models.
  • Specific curve types, like twisted Edwards, Huff, and quartic curves, are particularly conducive to long progressions.
  • These findings contribute to the understanding of rational points on elliptic curves and their properties.