What's the definition of basic algebra that makes that fact obvious? I took 3 semesters of algebra in college (Linear Algebra, Abstract Algebra I and II), and don't see anything obvious about it. But then, neither does Terry Tao:

> "the powers φ, φ2, φ3, … of the golden ratio lie unexpectedly close to integers...

It's not basic algebra in the sense of my high-school "Maths B" class. But it is just the kind of thing we did in "Maths C".

Here a hint: use the definition of φ that

  1 = φ - 1/φ

And use it to prove that if

  F[k] =  φ^k + 1/(-φ)^k

is an integer, then

  F[k+1] =  φ^(k+1) + 1/(-φ)^(k+1)

Must also be an integer. Now show that this proves by induction that all F[k] are integers and note what happens to 1/(-φ)^k as k grows to infinity.