5. Is the following conjecture true or false? Justify your conclusion.
Conjecture. Let $f_1, f_2, \dots, f_m, \dots$ be the sequence of the Fibonacci numbers. For each natural number $n$, the numbers $f_n f_{n+3}$, $2 f_{n+1} f_{n+2}$, and $(f_{n+1}^2 + f_{n+2}^2)$ form a Pythagorean triple.