The Sequence of Lucas Numbers Is Not Stable Modulo 2 and 5

Let L0 = 2, L1 = 1, and Ln = Ln−1 + Ln−2 for n ≥ 2, denote the sequence L of Lucas numbers. For any modulus m ≥ 2, and residue b (mod m), denote by vL(m, b) the number of occurrences of b as a residue in one (shortest) period of L modulo m. In this paper, we completely describe the functions vL(pk, .) for k ≥ 1 in the cases p = 2 and p = 5. Using a notion formally introduced by Carlip and Jacobson, our main results imply that L is neither stable modulo 2 nor modulo 5. This strikingly contrasts with the known stability of the classical Fibonacci sequence modulo these two primes.