Let's define sequence $S_i$ as :

$ S_i= S^4_{i-1}-4\cdot S^2_{i-1}+2 ~\text{with}~ S_0=8$

I have found that :

$F_2 \mid S_1 , ~F_3 \mid S_3 ,~F_4 \mid S_7 $

where $F_2 , F_3 , F_4 $ are Fermat numbers .

Conjecture :$ F_n = 2^{2^n}+1 ,(n \geq 2) ~\text{is a prime iff}~F_n \mid S_{2^{n-1}-1}$

In this document you can find my proof of this conjecture .

**Question :**

Is my proof acceptable ? Are there similar criteria in the literature ?