If $n\ge 2$ is an integer.
Can $N=2^n-1$ or $N=2^n+1$ be a weak 3-Fermat pseudoprime , in other words can $N$ satisfy $$3^{N-1}\equiv 1\mod N$$ and nevertheless be composite ?
For $2\le n \le 5\ 000$ , neither of the numbers have this property.
In particular : It is well known that a Fermat-number $$N:=2^{2^n}+1$$ is prime iff $$3^{(\frac{N-1}{2})}\equiv -1 \mod N$$ holds. Is $3^{N-1}\equiv 1\mod N$ also enough to prove the primality ?