Sorry, I couldn't get access to the paper you referred to. For a prime modulus $p>5$ it is easy to show that the period of the Fibonacci sequence is a factor of either $p-1$ or $2(p+1)$ depending on whether $5$ is a quadratic residue modulo $p$ or not (or by reciprocity, whether $5\equiv\pm1\pmod5$ p\equiv\pm1\pmod5$ or not). This is because the roots of $$ x^2-x-1=0 $$ over $F_p$ are either in $F_p^*$ (if that polynomial factors modulo $p$), or in $F_{p^2}^*$. In the former case they are roots of unity of order that is a factor of $p-1$. And in the latter case the roots $\tau_1,\tau_2$ are conjugates of each other, and hence satisfy both equations $$ \tau_1^p=\tau_2,\qquad \tau_1\tau_2=-1 $$ implying that they both are roots of unity of order dividing $2(p+1)$. The periodicity then follows from Binet's formula.
Oh. Your program is leaving the integer domain. Are you sure you are not getting any overflows/loss of accuracy? You would need accuracy of something like 20 significant digits to handle $F_{200}$. Why aren't you using the recursive code? Storing two last entries is enough!
