Editing Fermat pseudoprime (section)

== Euler–Jacobi pseudoprimes ==
{{main|Euler–Jacobi pseudoprime}}
Another approach is to use more refined notions of pseudoprimality, e.g. [[strong pseudoprime]]s or [[Euler–Jacobi pseudoprime]]s, for which there are no analogues of [[Carmichael number]]s. This leads to [[randomized algorithm|probabilistic algorithm]]s such as the [[Solovay–Strassen primality test]], the [[Baillie–PSW primality test]], and the [[Miller–Rabin primality test]], which produce what are known as [[industrial-grade primes]]. Industrial-grade primes are integers for which primality has not been "certified" (i.e. rigorously proven), but have undergone a test such as the Miller–Rabin test which has nonzero, but arbitrarily low, probability of failure.