Let the $f(x) = x^2 -ax+b$ has a positive discriminant $D=a^2-4b$ and $k,l$ be its roots.
Then $U_n = \frac{k^n-l^n}{k-l}$ and $V_n=k^n+l^n$.
I would like to prove these 4 properties
- If $U_n$ is a perfect square then $n=1,2,3,6$ or $12$
- If $V_n$ is a perfect square then $n=1,3$ or $5$.
- If $U_n$ is 2 x perfect square then $n=3$ or $6$
- If $V_n$ is 2 x perfectt square then $n=3$ or $6$
but i have no idea how. May anyone help?