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Old 2020-06-30, 13:01   #9
Dr Sardonicus
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Feb 2017

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Originally Posted by ATH View Post
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 Pari-GP 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 "K-Carmichael 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 In´Čünitely Many Perrin Pseudoprimes.

The ne plus ultra of "generalized Carmichael numbers" appears to be described in a paper entitled Higher-order Carmichael numbers by Everett W. Howe.
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