> what if you use the closed-form answer, but without any floating-point or other approximations You can use a field extension here, ℚ(√5), to represent your n

> what if you use the closed-form answer, but without any floating-point or other approximations You can use a field extension here, ℚ(√5), to represent your numbers. The numbers would be represented as pairs, (a,b), which represents the number a+b√5. You can see for yourself that this set is closed under addition and multiplication. But when you try to implement code that computes exponents with this representation, you’re probably going to fall back to the old “repeated squaring” trick. But that’s how you calculate exponents with your matrix representation anyway—and you can note that the matrix powers (for this matrix) always take the form | a b | | 0 a | So you can represent these matrixes as (a,b) pairs anyway. We are right back where we started—our state is represented as a pair of numbers, and we use repeated squaring to calculate an exponent. Sometimes these different ideas for representing the problem give us new insight or new algorithms, and sometimes it turns out to be a new perspective on the same piece of code.