Quote:
Originally Posted by ATH
Apparently the constructed set of primes where many normal BPSW counter examples should exist is a weaker version of BPSW. So they are no proofs that normal BPSW psp's exists?

I believe there is indeed no such proof. One possible type of proof would be a numerical example. It is however not the only possibility.
My old PariGP manual says that no examples of composite numbers which "pass" BPSW are known, but that it is thought that there are infinitely many such numbers.
I mention that, a number of years ago I conceived the notion of a generalization of Carmichael numbers that enlarged the scope of testing to the algebraic integers in a number field. I was unable to prove my suspicion that for a given number field K there are infinitely many "KCarmichael numbers," or that there were infinitely many Carmichael numbers composed of primes which "split completely" in K/Q. However, Jon Grantham subsequently succeeded in proving this, and kindly sent me a copy of the paper,
There are Inﬁnitely Many Perrin Pseudoprimes.
The
ne plus ultra of "generalized Carmichael numbers" appears to be described in a paper entitled
Higherorder Carmichael numbers by Everett W. Howe.