This is an overview of ABC triples for PRiME. LaTeX code will be used somewhat liberally! Hopefully one day we can put jsmath on Rhea so that things TeX in place. I'm not using the math tags, because they don't display well, and can't be copied directly into LaTeX files later.

History and Motivation

Historically, mankind has been fascinated with the art of determining integer solutions, and rational solutions to polynomial equations. An early master of this art was Diophantus of Alexandria, who lived sometime between 300 BC and 300 AD. In his honor, such equations, with the implied task of finding some (or all) integer (or rational) solutions are referred to as Diophantine equations.

Pythagorean Triples

The most famous diophantine equation is the one which arises from the Pythagorean theorem:

$$a^2 + b^2 = c^2.$$

Integer solutions (a,b,c) to this equation are called pythagorean triples. They were deeply studied in ancient times because of there usefulness in constructing right angled triangles whose sides were all multiples of a common length.

Are there infinitely many Pythagorean triples?

Suppose that (a,b,c) is a Pythagorean triple. Then for any positive integer d, we have

$ (d a)^2 + (d b)^2 = d^2 ( a^2 + b^2) = d^2 c^2 = (d c)^2 $

So if we have one solution, say $ 1^2 + 0^2 = 1^2 $ we get infinitely many for free, but this trick is cheap -- and BORING -- we can actually do much better. We'll say that a Pythagorean triple (a,b,c) is primitive if gcd(a,b,c)=1 and $ abc \neq 0 $