A Perrin pseudoprime in the unrestricted sense is a composite number 
satisfying 
,
where 
is an element of the Perrin sequence 3, 0, 2,
3, 2, 5, 5, 7, 10, 12, 17, ... (OEIS A001608).
Every prime number 
satisfies 
. The smallest unrestricted Perrin pseudoprimes are 271441,
904631, 16532714, 24658561, ... (OEIS A013998).
Unlike a Perrin prime, a Perrin pseudoprime is a
composite index, rather than a prime number value
of the Perrin sequence.
Adams and Shanks (1982) discovered the smallest unrestricted Perrin pseudoprime after unsuccessful searches by Perrin (1899), Malo (1900), Escot (1901), and Jarden (1966). Stewart (1996) incorrectly stated that no Perrin pseudoprimes were then known.
Grantham (2001) defined a Perrin pseudoprime with integer parameters 
to be an odd composite
number 
for which either
1. 
and 
has an S- or I-recurrence relation signature,
or
2. 
and 
has a Q-recurrence relation signature,
where 
is the discriminant of 
and 
is the Jacobi symbol.
The original Perrin sequence corresponds to 
and 
. For these parameters, Grantham's restricted Perrin
pseudoprimes begin 27664033, 46672291, 102690901, ... (OEIS A275613).
The minimal restricted Perrin test imposes only 
and 
on a composite
number 
.
Its pseudoprimes begin 27664033, 46672291, 102690901, ... (OEIS A018187)
and need not satisfy Grantham's signature conditions. Kurtz et al. (1986)
found all 55 minimal restricted Perrin pseudoprimes less than 
, each of which has an S-recurrence
relation signature.
."
Math. Comput. 46, 691-701, 1986.Malo, E. "Réponse
à 1484." L'Intermédiare des Math. 7, 280-282 and
312-314, 1900.Perrin, R. "Item 1484." L'Intermédiare
des Math. 6, 76-77, 1899.Ribenboim, P. 