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Rational Points in Arithmetic Progression on the Unit Circle
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Ajai Choudhry

13/4 A Clay Square

Lucknow - 226001

India

Abhishek Juyal

Department of Mathematics

Motilal Nehru National Institute of Technology

Allahabad - 211004

India

**Abstract:**

Several authors have considered the problem of finding rational points
(*x*_{i}, y_{i}),
*i* = 1, 2,..., *n*
on various curves *f*(*x*, *y*) = 0, including
conics, elliptic curves and hyperelliptic curves, such that the
*x*-coordinates *x*_{i}, *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 *x*^{2} + *y*^{2} = 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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