True O(logn) (for 64bit numbers) and deterministic primality test using Fibonacci numbers

Revision en3, by CoolManGame, 2023-09-21 20:37:14

I unexpectedly found a simple primality condition when browsing the prime number sequence on OEIS. I thought it would be nice to share it here since I haven't seen it shared anywhere else. It's also easier to remember than the Miller-Rabin test. Here's some pseudocode to describe the primality test:

isPrime(N):
if N <= 10: return N in [2, 3, 5, 7]
return (Fib(N) % N == 1 or Fib(N) % N == N-1) and (2**(N-1) % N) == 1


(Note: (Fib(N) % N) and (2**(N-1) % N) can be evaluated in O(logn))

This does work, and I've gotten AC on https://www.spoj.com/problems/PON/ with it.

You can find this conjecture on https://oeis.org/A000040 where it said "Conjecture: Sequence = {5 and n <> 5| ( Fibonacci(n) mod n = 1 or Fibonacci(n) mod n = n — 1) and 2^(n-1) mod n = 1}. — Gary Detlefs, May 25 2014". It's a conjecture, but conjectures are true in CP am I right lol.

I haven't found a name for it, nor have I seen it shared anywhere else. Though, if it's already known with a published name (I'm not the best googler lol), do let me know through the comments or DM.

#### History

Revisions

Rev. Lang. By When Δ Comment
en4 CoolManGame 2023-09-22 13:37:08 335 Making it known that the conjecture is false
en3 CoolManGame 2023-09-21 20:37:14 0 (published)
en2 CoolManGame 2023-09-21 20:36:00 6 Reformatted code block (saved to drafts)
en1 CoolManGame 2023-09-21 20:33:52 1184 Initial revision (published)