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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 = m2n2

                    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

For m =  and  n =  the Pythagoras's triple is

a = m2−n2 =       b = 2mn =      c = m2+n2 = 

 

 

Exercise

Find the Pythagoras's triple with m = and n = in Euler's formula:

a =      b =      c = 

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

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