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Pythagoras Theorem

Pythagoras's Triples

A triple (a, b, c) of integers that satisfies the equations

                    a² + b² = c²

is called a Pythagoras’ triple. There are infinitely many such triples.

For example, (3, 4, 5), (5, 12, 13), (8, 15, 17) are Pythagoras’ triples.

Here is one method (Euler) for generating Pythagoras’ triples:

For any positive integers m > n, let

                    a = m² − n²

                    b = 2mn

                    c = m² + n²

Then (a, b, c) is a Pythagoras’s triple.

For example, if n = 1 and m = 2 then

                    a = 3, b = 4, c = 5 and

                    3² + 4² = 5²

Example

Exercise

On the website http://www.cut-the-knot.org/pythagoras/pythTriple.shtml Alexander Bogomolny provides an applet that generates all Pythagoras’s triples.

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