To pick a little further at this nit; bigint is no good, right? The only way to compute phi^n with bigints is to do an O(log n) operations in the algebraic int

To pick a little further at this nit; bigint is no good, right? The only way to compute phi^n with bigints is to do an O(log n) operations in the algebraic integers Z(5) (similar to exponentiating complex integers). Doing it with floats/doubles is a little more reasonable for an O(1) approach, but even that approach breaks down for large enough n.