The USSR Olympiad Problem Book
#178.
*
Prove that if \(x_1\) and \(x_2\) are roots of the equation \(x^2 - 6x + 1 = 0\), then \(x_1^n + x_2^n\) is, for any natural number \(n\), an integer not divisible by 5.
Hint.
Fibonacci!