# Pythagorean triples

In a right triangle with sides $a@ and $b@ and hypotenuse
$c@ we have
$$ a^{2} + b^{2} = c^{2} . @@
If $a@, $b@, and $c@ are all positive integers, this is called
a **Pythagorean triple**.
The best known example, and the simplest one, is $3@, $4@, $5@ since
$$ 9 + 16 = 25 @@

and another
well known one is
$5@, $12@, $13@.
One of the very
earliest mathematical documents we know of
is concerned with Pythagorean triples - it is a tablet
from Babylon dated to roughly 1800 B.C.
(This was a long time ago. But already by then cuneiform or
its pictographic predecessor had been in use for at least 1200 years,
a time span equal to that since Charlemagne was crowned emperor!)

If $(a, b, c)@ is a Pythagorean triple, then
the point $(a/c, b/c)@ is a point on the unit circle
with rational coordinates. In fact, finding out all Pythagorean triples
turns out to be equivalent to
finding all such points. This allows us to describe easily
a parametrization of Pythagorean triples that may have been discovered
by the Babylonians, according to legend was discovered also
by Pythagoras himself.
This parametrization was known in some form to Euclid
(Lemma 1 to Proposition 29 of Book X).
It was also used implicitly by Diophantus, and characterized
fully by Leonardo of Pisa, conceivably the very first
mathematician of Western Europe, about 1200 CE.

The parametrization left open the question of exactly
which integers could be the hypotenuse in a Pythagorean triple,
or equivalently: *Which integers can be expressed
as the sum of two squares*? This was answered by Fermat
in the early 17th century,
and several proofs have been found since, by
Legendre and Gauss among others.
Pythagoren triples have thus occupied mathematicians
for more than 3500 years. And the story is not yet
finished - at least
one relevant question has not yet been answered.

## References

- O. Neugebauer,
**The exact sciences in antiquity**, Dover, 1969.
This is a reprint of the 1957 second edition. Chapter 2 spends a great
deal of effort on the tablet.

Written by Bill Casselman.