Rational points on the unit circle

Is anything known about any of the following questions about rational points on the unit circle? By “double point” I mean an element of 2C, where C is the group of rational points on the unit circle (i.e. Gaussian rationals with unit modulus, under multiplication).

  1. Are there two rational points (x1,y1) and (x2,y2) on the unit circle such that the numbers y1+y2 and y1y2 are also the y-coordinates of rational points on the unit circle, except where y1 or y2 is zero?

  2. Are there are three distinct double points on the unit circle whose y-coordinates are in arithmetic progression, other than the trivial case where the middle one is on the x-axis?

  3. Are there two double points (x1,y1) and (x2,y2) on the unit circle such that y1+y2 is also the y-coordinate of a double point on the unit circle, excluding the trivial case where one of y1, y2, y1+y2 is zero?

  4. Are there three distinct rational points on the unit circle whose y-coordinates are in arithmetic progression, where the common difference of this arithmetic progression is also the y-coordinate of a rational point on the unit circle, other than the trivial case where the middle point is on the x-axis?

Answer

Attribution
Source : Link , Question Author : Robin Houston , Answer Author : Community

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