Journal of Integer Sequences, Vol. 19 (2016), Article 16.4.1

Rational Points in Arithmetic Progression on the Unit Circle

Ajai Choudhry
13/4 A Clay Square
Lucknow - 226001

Abhishek Juyal
Department of Mathematics
Motilal Nehru National Institute of Technology
Allahabad - 211004


Several authors have considered the problem of finding rational points (xi, yi), i = 1, 2,..., n on various curves f(x, y) = 0, including conics, elliptic curves and hyperelliptic curves, such that the x-coordinates xi, i = 1, 2,..., n are in arithmetic progression. In this paper we find infinitely many sets of three points, in parametric terms, on the unit circle x2 + y2 = 1 such that the x-coordinates of the three points are in arithmetic progression. It is an open problem whether there exist four rational points on the unit circle such that their x-coordinates are in arithmetic progression.

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Received December 10 2015; revised version received March 14 2016. Published in Journal of Integer Sequences, April 7 2016.

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