 
Summary: The Modular Tree of Pythagoras
Roger C. Alperin
1. INTRODUCTION. The Pythagorean triples [x, y, z] of integers satisfying the
equation x2
+ y2
= z2
have been studied and enumerated since Babylonian times.
Since Diophantus, it has been known that this set of triples is related to the standard
rational parameterization of the circle of radius one, namely,
t
t2
 1
t2 + 1
,
2t
t2 + 1
.
The Pythagorean triples that are relatively prime (called the primitive triples) have
the elementary and beautiful characterization as integers x = m2
 n2
