Define "probable prime" (PP) to be natural $n>1$ satisfying $2^{n-1} \equiv 1 \pmod{n}$ or $n=2$.

Probable primes are the union of the primes and base two pseudoprimes.

This definition is much simpler than the definition for primes and the primes are sufficiently large subset of probable primes.

Are there open problems for primes which are known for probable primes?

Positive answer doesn't necessarily mean the problem is solved for the primes (e.g. infinitely many twin PP hypothetically might mean finitely many twin primes and infinitely many twin base 2 pseudoprimes).