Prove without induction $2^n \mid (b+\sqrt{b^2-4c})^n + (b-\sqrt{b^2-4c})^n $ 
Prove $2^n \mid (b+\sqrt{b^2-4c})^n + (b-\sqrt{b^2-4c})^n $ for all $n\ge 1$ and $b,c$ are integers.


Is it possible to prove this without induction? 

The sequence given by:
$$ a_n = \left(b+\sqrt{b^2-4c}\right)^n + \left(b-\sqrt{b^2-4c}\right)^n $$
satisfies the recurrence relation:
$$ a_{n+2} = \color{red}{2}b\cdot a_{n+1} - \color{red}{4}c\cdot a_{n}.$$
Since $\nu_2(a_n)\geq n$ holds for $n=0$ and $n=1$, it holds for every $n$ by the previous relation.
Ok, this is still induction :D
