# 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.