Summary: 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.

11D09

arithmetic progression, unit circle