Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.6

Arithmetic Progressions on Conics

Abdoul Aziz Ciss
Laboratoire de Traitement de l'Information et Systèmes Intelligents
École Polytechnique de Thiès
BP A10 Thiès

Dustin Moody
Computer Security Division
National Institute of Standards and Technology (NIST)
100 Bureau Drive
Gaithersburg, MD 20899-8930


In this paper, we look at long arithmetic progressions on conics. By an arithmetic progression on a curve, we mean the existence of rational points on the curve whose x-coordinates are in arithmetic progression. We revisit arithmetic progressions on the unit circle, constructing 3-term progressions of points in the first quadrant containing an arbitrary rational point on the unit circle. We also provide infinite families of 3-term progressions on the unit hyperbola, as well as conics ax2 + cy2 = 1 containing arithmetic progressions as long as 8 terms.

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Received June 18 2016; revised versions received September 13 2016; November 15 2016. Published in Journal of Integer Sequences, December 27 2016.

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