The ABC Conjecture and its Consequences on Curves

We begin with a problem that most high school students have seen the solution to: what triples of integers represent the sides of a right triangle? The Pythagorean equation for a triangle with sides x, y, and z says that the sum of the squares of the two legs must equal to the square of the hypotenuse. By Pythagoras’s theorem, the problem of finding right triangles with integer sides becomes the problem of finding integral points on the projective curve x2 + y2 = z2. Since the solutions to Pythagoras’s equation are homogenous we can make the substitutions X = x/z and Y = y/z to obtain the new equation X2 + Y 2 = 1. Solutions to x2 + y2 = z2 in integers correspond to rational solutions of X2 + Y 2 = 1. However, X2 + Y 2 = 1 is a familiar geometric object: it defines the unit circle in R2. By projecting from the point (0, 1) we can parametrize the unit circle. A line of slope −t starting at (0, 1) intersects the circle at one other point, ( 2t 1 + t2 , t2 − 1 1 + t2 ) .