This is an overview of ABC triples for PRiME.

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 given by the pythagorean theorem:

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

Solutions to this equation are called pythagorean triples. They were deeply studied in ancient times.

What are some Pythagorean Triples?

Most people know that (3,4,5) is a Pythagorean triple, that is,

$ 3^2 + 4^2 = 5^2. $