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## 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 Pythagoras’s triples.