How to make primality test on paper without any calculator for high numbers like $$ (32^{200}) - 1 $$
$$ (400^{555}) - 1 $$ What specific test is useful in such cases: $$(a^b) - 1$$ where a and b are high numbers), Fermat's theorem, AKS or is there any other test that shows that this equation is prime number?
$a-1$ is a factor of $a^b-1$. So if $a>2$, then $a^b-1$ is composite. For the case of $a=2$, you are looking for Mersenne primes.