Let us define the sequences $a_1, a_2, a_3,...$ and $b_1, b_2, b_3,...$. with the following conditions $a_1 = 3, b_1 = 1$ and $a_{n +1} =\frac{a_n^2+b_n^2}{2}$ and $b_{n + 1}= a_n \cdot b_n$ for each $n = 1, 2,...$. Find all different prime factors οf the number $a_{2000} + b_{2000}$.