Geodesic
Rational points in arithmetic progression on the unit circle
Journal of integer sequences, Tome 19 (2016) no. 4
Several authors have considered the problem of finding rational points $(x_{i}, y_{i}), i = 1, 2,\dots , n$ on various curves $f(x, y) = 0$, including conics, elliptic curves and hyperelliptic curves, such that the $x$-coordinates $x_{i}, i = 1, 2,\dots , n$ 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.
Classification : 11D09
Keywords: arithmetic progression, unit circle 
@article{JIS_2016__19_4_a6,
     author = {Choudhry,  Ajai and Juyal,  Abhishek},
     title = {Rational points in arithmetic progression on the unit circle},
     journal = {Journal of integer sequences},
     year = {2016},
     volume = {19},
     number = {4},
     zbl = {1415.11060},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2016__19_4_a6/}
}
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Choudhry,  Ajai; Juyal,  Abhishek. Rational points in arithmetic progression on the unit circle. Journal of integer sequences, Tome 19 (2016) no. 4. http://geodesic.mathdoc.fr/item/JIS_2016__19_4_a6/