The Fermat primality test base $a$ says $N$ is probably prime if $a^{N-1}=1\mod N$. $N$ is a pseudoprime base $a$ if it is probably prime, but not prime.
Does there exist a base $a$ such that the only pseudoprimes base $a$ are Carmichael numbers?
More generally, could we find $k$ bases such that if $N$ is a pseudoprime to all of these bases, then $N$ is a Carmichael number?
To solve this I think we're gonna construct a pseudoprime base $a$ then show its not a Carmichael number by the classification theorem. Cant find any reference for this.