Let $\displaystyle F_{0}, F_{1}, F_{2}, F_{3},...$ be the Fibonacci numbers, defined by

$\displaystyle F_{0}=1$

$\displaystyle F_{1}=1$

$\displaystyle F_{n}=F_{n-1}+F_{n-2}$

for $\displaystyle n\geq 0$

a) Prove that for all $\displaystyle n\geq 0$ we have $\displaystyle gcd(F_{n+1}, F_{n})=1$

b) Prove that for all $\displaystyle n\geq 0$ we also have $\displaystyle gcd(F_{n+2}, F_{n})=1$

(Check what Euclid's algorithm would do if you started to compute $\displaystyle gcd(F_{n+1}, F_{n})$ or $\displaystyle gcd(F_{n+2}, F_{n})$.)

Any help would be greatly appreciated!