No fibonacci number is congruent to 4 or 6 modulo 8. They cycle [0,1,1,2,3,5,0,5,5,2,7,1]
. So modulo 1000 there are numbers you can't reach.
More info:
$$\begin{align}
\text{modulus} && \text{Fibonacci period} && \text{# of missing residues} \\
2 && 3 && 0\\
2^2 && 6 && 0\\
2^3 && 12 && 2\\
2^5 && 48 && 11 = 2^5 \cdot \frac{11}{32}\\
2^8 && 3 \cdot 2^7 && 88 = 2^8 \cdot \frac{11}{32} \\
2^{15} && 3 \cdot 2^{14} && 5637 = 2^{15} \cdot \frac{11}{32} \\
2^{16} && 3 \cdot 2^{15} && 11264 = 2^{16}\cdot \frac{11}{32} \\
2^{20} && 3 \cdot 2^{19} && 360448 = 2^{20} \cdot \frac{11}{32} \\
5 && 20 && 0 \\
5^2 && 100 && 0 \\
5^3 && 500 && 0 \\
5^4 && 2500 && 0 \\
10 && 60 && 0 \\
10^2 && 300 && 0 \\
10^3 && 1500 && 250 \\
10^4 && 15000 && 3125 = 10^4 \cdot \frac{5}{16}\\
10^5 && 150000 && 34375 = 10^5 \cdot \frac{11}{32}\\
10^6 && 1500000 && 343750 = 10^6 \cdot \frac{11}{32}\\
\end{align}$$
Conjectures:
Modulo $2^n$, the period is $3 \cdot 2^{n-1}$. For $n \ge 5$, exactly $\frac{11}{32}$ of the residues will not be present in the cycle.
Modulo $5^n$, the period is $4 \cdot 5^n$. All residues are present.
Modulo $10^n$, for $n \ge 3$, the period is $15 \cdot 10^{n-1}$. For $n \ge 5$, exactly $\frac{11}{32}$ of the residues will be missing.