The sequence of Lucas numbers is not stable modulo $2$ and $5$

Let $L_0=2, L_1=1$, and $L_n=L_{n-1}+L_{n-2}$ for $n \ge2$, denote the sequence $\mathcal{L}$ of Lucas numbers. For any modulus $m\ge2$, and residue $b\pmod m$, denote by $v_\mathcal{L}(m,b)$ the number of occurrences of $b$ as a residue in one (shortest) period of $\mathcal{L}$ modulo $m$. In this paper, we completely describe the functions $v_\mathcal{L}(p^k,.)$ for $k \ge 1$ in the cases $p=2$ and $p=5$. Using a notion formally introduced by Carlip and Jacobson, our main results imply that $\mathcal{L}$ is neither stable modulo $2$ nor modulo $5$. This strikingly contrasts with the known stability of the classical Fibonacci sequence modulo these two primes.