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 probably prime with respect to the polynomial .

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

REFERENCES:

Grantham, J. "Frobenius Pseudoprimes." 1996. https://www.pseudoprime.com/pseudo1.ps.

Grantham, J. "A Frobenius Probable Prime Test with High Confidence." 1997. https://www.pseudoprime.com/pseudo2.ps.

Grantham, J. "Pseudoprimes/Probable Primes." https://www.pseudoprime.com/pseudo.html.