Is the BPSW-test always correct for special kinds of numbers? Here
https://en.wikipedia.org/wiki/Baillie%E2%80%93PSW_primality_test
a powerful primality test is mentioned for which no counterexample is known and it is claimed that none upto $2^{64}$ exists.

Can we prove that the test never fails for particular kind of numbers, Fermat numbers , Mersenne numbers , numbers of the form $n^2+1$ , $\cdots$ ?

Any reference would be welcome.

On this page of Thomas R. Nicely there is a lot of information and also the source of the claim.
https://faculty.lynchburg.edu/~nicely/misc/bpsw.html
