The "compute the eigenvalues of Matrix([[1,1],[1,0]]) raised to the power N" approach to computing the Nth Fibonacci number is only constant time if you assume exponentiation is constant time, which it is not on any computer or realistic model of computation.

I still think this is a really cool technique though, even if it only works in "math land." http://bit.ly/fib_using_eigenvals

Constant time fibonacci is derived as an example for linear basis change in the superb book "Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach".

It's all contained in the sample pages http://matrixeditions.com/VC5.Chap2.219-221.pdf

However, even though the formula is O(1), the runtime of the algorithm might be larger.

Simply because a standard 64bit integer might be too small. And the BigNum implementation needs more time to multiply all the bits.

There is no constant time implementation. Binet's formula uses exponentiation, which gives us log(n) time.

No, but I go into it a bit in https://lee-phillips.org/lispmath/

O(1)? Having all the answers pre-cached would be the only way to do that.

Unless there is some kind of quantum computer algorithm that fits this problem well.