Pythagore Triple

A Pythagorean triplet is a set of three natural numbers $ a $, $ b $ and $ c $ such that $ a^2+b^2=c^2 $

Example: (3,4,5) is a triplet of Pythagoras because $ 3^2+4^2=5^2 $

It exists heuristics to find a Pythagore Triple but the easiest method consists in testing iteratively all possibilities of a and b when s is given, the value of c is constrained by s=a+b+c.

The following equations can be deducted:

$$ a^2 + b^2 = (s − a − b)^2 \\ a <= (s − 3)/3 \\ b < (s − a)/2 $$

Example: If $ s = 12 $, then $ a <= 3 $ and $ b < 4.5 $, a quick test allows to find $ a = 3, b = 4 $ and get the triple $ \{3,4,5\} $.

The first Pythagorean triples (side inferior to 100)

There is no Pythagorean triplet with 2 identical values. Indeed if 2 sides are $ a $ (natural integer), the last side is $ a \sqrt2 $ which can not be an integer.

Example: $ a = 1 $ the triplet becomes $ (1, 1, \sqrt2) $. By scaling, it is not possible to obtain an both isosceles and right triangle with integer sides.