Golden powers are nearly integers
Specifically, they're nearly Lucas numbers.  For those unfamiliar, the Lucas numbers are defined by the recursion L_0 = 2, L_1 = 1, and L_n = L_{n-1} + L_n.  (I.e., the Fibonacci recursion, but with different starting values.)  They can also be given explicitly by L_n = phi^n + (-1/phi)^n.  Since (-1/phi)^n goes rapidly to zero, phi^n will be very close to L_n.  More specifically, for n>=2, (1/phi)^n < 1/2, so L_n will be the nearest integer to phi^n.<p>The Fibonacci numbers, meanwhile, can be given by the formula F_n = (phi^n - (-1/phi)^n) / sqrt(5).  So a similar thing holds for them.<p>More generally, phi is a Pisot number: <a href="https://en.wikipedia.org/wiki/Pisot%E2%80%93Vijayaraghavan_number" rel="nofollow">https://en.wikipedia.org/wiki/Pisot%E2%80%93Vijayaraghavan_n...</a>  Any Pisot number will have its powers approach integers at an exponential rate.  (In that, the distance to the nearest integer decreases exponentially.)