Fibonacci goes magic (Q2341516)

Let \(P = \left ( \begin{matrix} 0 \;\;1 \\ 1 \;\;1 \end{matrix} \right )\), \(\mathrm{Id}\) be an identity matrix, \(\gamma = \gamma (p)\) be the smallest integer \(m\) such that \(P^m = \mathrm{Id} \pmod p\). The prime \(p\) for which \(P^{\gamma/2} = -\mathrm{Id}\) is called a \textit{good prime}. Let \((a, b) \neq (0, 0)\) and \(x_0 = a, x_1 = b, x_n = x_{n-1} + x_{n-2} \pmod p\) for \(n \geq 2\). The good prime \(p\) is called a \textit{very good prime} if for every choice of \((a, b) \neq (0, 0)\), the associated sequence contains \(\nu\) zeros. Some properties of the good and very good primes are studied.