## Pythagorean triangle – area

The integers:

$a \; = \; u^2 \; - \; v^2$
$b \; = \; 2 \, u \, v$
$c \; = \; u^2 \; + \; v^2$

form a Pythagorean triple.

where   $u, \; v$   are integers with   $u \; > \; v \; > \; 0$

$a, \; b, \; c$   are the lengths of the sides of a right triangle.

Prove that the area of such a triangle is not a perfect square when

$u = F_{n+1}$ ,     $v = F_{n}$ ,   and   $n \; \geq \; 2$

$F_{n}$   is the n-th Fibonacci number