DISTRIBUTION OF FIBONACCI AND LUCAS NUMBERS MODULO 3k

Let F0 = 0, F1 = 1, and Fn = Fn−1 + Fn−2 for n ≥ 2 denote the sequence F of Fibonacci numbers. For any modulus m ≥ 2 and residue b (modm), denote by vF (m, b) the number of occurrences of b as a residue in one (shortest) period of F modulo m. Moreover, let vL(m, b) be similarly defined for the Lucas sequence L satisfying L0 = 2, L1 = 1, and Ln = Ln−1 + Ln−2 for n ≥ 2. In this paper, completing the recent partial work of Shiu and Chu we entirely describe the functions vF (3 , .) and vL(3 , .) for every positive integer k. Using a notion formally introduced by Carlip and Jacobson, our main results imply that neither F nor L is stable modulo 3. Moreover, in terms of another notion introduced by Somer and Carlip, we observe that L is a multiple of a translation of F modulo 3 (and conversely) for every k.