I’m interested in some sequences which have no square factor. $$ s_i =\begin{cases} 4 & i=0; \\\\ s_{i-1}^2 - 2 & \text{otherwise} \end{cases}$$ This is Lucas–Lehmer primality test sequence. A003010 in OEIS.
when ${s_i}$ has a square factor? I had checked that the sequence has no square factor for i=1,2,...,7