## Pythagorean triples and rational points on plane conic sections

**Goal:**

Pythagorean triples are integer triples (a,b,c) occurring as side lengths of
rectangular triangles. By Pythagoras' Theorem, they are hence the triples such
that a^2 + b^2 = c^2.
Setting x=a/c and y=b/c we obtain x^2 + y^2 = 1. Thus, our Pythagorean triple
has given rise to a rational point on the unit circle; the word "rational"
refers to the fact that the coordinates (x,y) are rational
numbers. Conversely, any rational point on the unit circle gives rise to a
"primitive" Pythagorean triple (that is, a triple such that the gcd of a,b,c
is 1). Any Pythagorean triple is proportional to a primitive one, so it is no
loss just to consider primitive triples.
Summary: Primitive Pythagorean triples correspond precisely to rational points
on the unit circle.

As a generalisation, one can consider rational points on other (smooth) plane
curves. The circle is a special case of a plane conic section.
Rational points on plane conic sections (assumed to be defined over Q) can be parametrised in a simple way:
fix a point on it and draw a line of rational slope through it; it will
intersect the conic section in one point.

Pythagorean triples can be enumerated; one way to obtain an enumeration is by
using the geometric description. Another one uses certain matrix groups.

The goal is to illustrate rational points on different curves, mostly plane
conic sections, and to give them a number theoretic meaning.
The subject can be developed into many different ways; this will be subject to
a discussion with interested participantes.

**Schedule:** To be determined.

**Supervisors:**
Gabor Wiese, Panagiotis Tsaknias

**Difficulty level:**
Introductory.

**Tools:**
Any computer language.

**Literature:**

- Alperin: The modular tree of Pythagoras
- wikipedia

**Results:**
[too be completed at the end of the project.]