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Arithmetic Progressions on Conics.

Abdoul Aziz Ciss1, Dustin Moody2

  • 1Laboratoire de Traitement de l'Information et Systèmes Intelligents, École Polytechnique de Thiès, BP A10 Thiès, Sénégal.

Journal of Integer Sequences
|August 4, 2017
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Summary
This summary is machine-generated.

This study explores arithmetic progressions on curves, finding up to 8-term progressions on specific conics like the unit hyperbola. Researchers constructed new 3-term progressions on the unit circle.

Keywords:
arithemetic progressionconic

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Area of Science:

  • Number Theory
  • Algebraic Geometry

Background:

  • Arithmetic progressions are sequences of numbers with a constant difference.
  • Rational points on curves are points with rational coordinates.
  • Previous work has explored arithmetic progressions on various curves.

Purpose of the Study:

  • To investigate the existence and construction of long arithmetic progressions on conics.
  • To extend the understanding of arithmetic progressions on the unit circle and unit hyperbola.
  • To generalize findings to a broader class of conics defined by ax^2 + cy^2 = 1.

Main Methods:

  • Utilizing the definition of rational points on curves.
  • Constructing specific examples of arithmetic progressions.
  • Developing general families of progressions for different conic sections.

Main Results:

  • Revisited and constructed 3-term arithmetic progressions on the unit circle, including those with an arbitrary rational point.
  • Provided infinite families of 3-term arithmetic progressions on the unit hyperbola.
  • Identified conics of the form ax^2 + cy^2 = 1 that contain arithmetic progressions of up to 8 terms.

Conclusions:

  • Arithmetic progressions can exist on various conics, with lengths depending on the specific curve.
  • The methods used allow for the construction of such progressions, offering insights into their structure.