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? 

Let $x_1$ and $x_2$ be the roots of the quadratic equation $$x^2 - bx + c = 0$$ 
Then $$x_1 = \frac{b + \sqrt{b^2 - 4c}}{2}$$ $$x_2 = \frac{b - \sqrt{b^2 - 4c}}{2} $$
The above statement is equivalent to $$2^n \,\big| \, 2^n\big({x_1}^n + {x_2}^n \big) $$
