000RM.dvi

210 Pythagorean triples

6.1 Primitive Pythagorean triples

It is well known that every primitive Pythagorean triple(a, b, c)is of the
form
a=m^2 −n^2 ,b=2mn, c=m^2 +n^2

for relatively prime integersmandnof different parity.

B

A 2 mn C

m m^2 −n^2

2 +

n^2

Some basic properties of Pythagorean triples:

1.Exactly one leg is even.

2.Exactly one leg is divisible by 3.

3.Exactly one side is divisible by 5.

4.The area is divisible by 6. Fermat has proved that the area of

a Pythagorean triangle can never be a square. Indeed, there is

no Pythagorean triangle with two sides whose lengths are square

(numbers).