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# Rational points on the unit circle

A rational point on a plane curve is a point on the curve with rational coordinates. For example, is a rational point on the circle with equation .

As one can guess from the example just given, rational points on are closely related to Pythagorean triples, i.e., the positive integer solutions to . In fact, if , , are any integers satisfying and , then will be a rational point on .

Conversely, if is a rational point on . then by choosing a common denominator for and one can write and for some integers with , and the relation implies . If moreover and are nonzero, then will all be nonzero, and will be a Pythagorean triple.

It would be nice to have a description of all the rational points on , because then we would have a description of all the Pythagorean triples. Our goal now is to find such a description using geometry!

Consider the following construction. Start with the rational point on . Fix a rational number . Draw the line with slope passing through . This line will intersect the circle at a second point (which depends on the number ).

By pure thought'' (no calculation), one can see that must have rational coordinates, because its -coordinate will arise as the solution to a quadratic equation which already has one rational root, namely the -coordinate of , and then the -coordinate of also will be rational (either by the same argument with -coordinates, or by using the equation of ).

For the incredulous, here is a full calculation of . The equation of (in point-slope form) is . To intersect this with , which is , substitute to obtain

It was not just luck that the quadratic polynomial in factored: the point is that it had to have as a factor, because we already knew that there was a point with -coordinate in the intersection , namely . Anyway, discarding the root and solving for the other possible -coordinate, we see that has -coordinate , and -coordinate

so

Since is rational, has rational coordinates.

We now claim that every rational point on the circle other than arises as for exactly one rational number . In other words, we obtain a parameterization of all the rational points on (except ). Recall that denotes the set of all rational numbers.

Theorem 1   The map

 rational points on other than

is a bijection (one-to-one correspondence).

Proof. There is a natural candidate for the inverse map, namely, the map

 rational points on other than

sending a rational point on other than to the slope of the line .

To show that the two maps are indeed inverse bijections, it suffices to show that the composition of the two maps in either order is the identity map.

Given , if we construct , and then take the slope of the line , we get back, by definition of .

On the other hand, if we start with a rational point on , compute the slope of the line , and then construct , then because is the intersection point other than of with the line through with slope . This completes the proof.

If and are positive integers with , and we take , then we obtain the point

so is a Pythagorean triple.

Next: Elliptic curves and Bezout's Up: Elliptic curves Previous: Plane curves of low
Zvezdelina Stankova-Frenkel 2001-09-22