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.