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8 Introduction to Rational Points on Plane Curves

(−1, 0) and (0, t) on the y-axis has equation y = t(x + 1).Iftheline L t intersects
2
2
the circle x + y = 1 at the points (−1, 0) and (x, y),thenwehave
1 − t 2 2t
x = and y = .
1 + t 2 1 + t 2
Observe that t is rational if and only if (x, y) is a rational point on the circle. The
value infinity corresponds to the base of the projection (−1, 0).

In order to prove the theorem of Diophantus, we consider for any primitive
Pythagorean triple (a, b, c) the number t = m/n, reduced to lowest terms, giving
the point on the y-axis corresponding to the rational point (a/c, b/c) on the circle
2
2
x + y = 1. The above formulas yield the relations
2 2 2 2
a = n − m , b = 2mn, c = n + m .
The first assertion of the theorem follows from the computation

2 2
2 2 2 2 2
n − m + (2mn) = n + m .
The above projection of the circle on the y-axis is also related to the following
trigonometric identities, left to the reader as an exercise,

θ

2 θ
θ sin θ 1 − tan 2 2 tan 2
tan = , cos θ = , sin θ = .
2 1 + cos θ 1 + tan 2 θ 1 + tan 2 θ
2 2

If R(sin θ, cos θ, tan θ, cot θ, sec θ, csc θ) dθ is an integral whose integrand is a ra-
tional function R of the six trigonometric functions, then it transforms into an integral

of the form S(t) dt, where S(t) is a rational function of t under the substitution
t = tan(θ/2). These classical substitutions of calculus come from the previous cor-
2
2
respondence between points on the y-axis and on the unit circle x + y = 1.
There is a natural generalization of the unit circle.
(3.2) Definition. The Fermat curve F n of order n is given by the equation in affine
x, y-coordinates