Let a sequence be defined by
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
Also define the associated polynomial
 |
(5)
|
and let 
be its discriminant. The Perrin sequence is a
special case corresponding to 
. The signature mod 
of an integer 
with respect to the sequence 
is then defined as the 6-tuple (
, 
, 
, 
, 
, 
) (mod 
).
1. An integer 
has an S-signature if its signature (mod 
) is (
, 
, 
, 
, 
, 
).
2. An integer 
has a Q-signature if its signature (mod 
) is congruent to (
) where, for some integer 
with 
,
,
,
and 
.
3. An integer 
has an I-signature if its signature (mod 
) is congruent to (
), where 
and 
.
