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.