Frobenius Pseudoprime

Let be a Monic Polynomial of degree with discriminant . Then an Odd Integer with is called a Frobenius pseudoprime with respect to if it passes a certain algorithm given by Grantham(1996). A Frobenius pseudoprime with respect to a Polynomial is then a composite Frobenius probablyprime with respect to the Polynomial .

While 323 is the first Lucas Pseudoprime with respect to the Fibonacci polynomial , the first Frobeniuspseudoprime is 5777. If , then any Frobenius pseudoprime with respect to is also a PerrinPseudoprime. Grantham (1997) gives a test based on Frobenius pseudoprimes which is passed by CompositeNumbers with probability at most 1/7710.

References

Grantham, J. ``Frobenius Pseudoprimes.'' 1996. http://www.clark.net/pub/grantham/pseudo/pseudo1.ps

Grantham, J. ``A Frobenius Probable Prime Test with High Confidence.'' 1997. http://www.clark.net/pub/grantham/pseudo/pseudo2.ps

Grantham, J. ``Pseudoprimes/Probable Primes.'' http://www.clark.net/pub/grantham/pseudo/.

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