Even the normal BPSW test has no known counterexamples and now they add a new test with only 5 vpsp under 10
^{15}. There must be only finitely many composites passing this stronger version and maybe none, I wonder if something like this will ever be proved.
Regarding the normal BPSW test:
https://mathworld.wolfram.com/Bailli...alityTest.html
Quote:
However, the elliptic curve primality proving program PRIMO checks all intermediate probable primes with this test, and if any were composite, the certification would necessarily have failed. Based on the fact that this has not occurred in three years of usage, PRIMO author M. Martin estimates that there is no composite less than about 10000 digits that can fool this test.

https://math.stackexchange.com/quest...tooopti?rq=1 See the bottom answer "
Determinism to 2^{64}"
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?
https://en.wikipedia.org/wiki/Bailli...primality_test
Quote:
However, a heuristic argument by Pomerance suggests that there are infinitely many counterexamples.[5] Moreover, Chen and Greene [6] [7] have constructed a set S of 1248 primes such that, among the nearly 2^{1248} products of distinct primes in S, there may be about 740 counterexamples. However, they are talking about the weaker PSW test that substitutes a Fibonacci test for the Lucas one.
