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 ![f(x) in Z[x]](/images/equations/FrobeniusPseudoprime/Inline7.svg)
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.
