I try some new seeds and I found this :

Let Wq=(2^q+1)/3, S0=q^2, and: S(i+1)=Si² (mod Wq)

Wq is a prime iff: Sq−1 ≡ S0 (mod Wq)

I tried until p<1000 and I found only Wagstaff prime

I used this code on Paridroid :

T(q)={Wq=(2^q+1)/3;S0=q^2;S=S0;print("q= ",q);for(i=1,q-1,S=Mod(S^2,Wq));if(S==S0,print("prime"))}

forprime(n=3,1000,T(n))

I don't know if the "-2" in the iteration part is important becauses it vanishes here but it seems it works with Mersenne numbers Mq=(2^q-1) too

And I tried with Fq=(3^q-1)/2 with S0=q³ and S(i+1)=Si³ and I found this

https://oeis.org/A028491 for the prime numbers

Maybe we can extend this for (n^q-1)/(n-1) and (n^q+1)/(n+1) with S0=q^n and S(i+1)=Si^n but I don't know