<s>
A	O
pseudoprime	B-Algorithm
is	O
a	O
probable	B-Algorithm
prime	I-Algorithm
(	O
an	O
integer	O
that	O
shares	O
a	O
property	O
common	O
to	O
all	O
prime	O
numbers	O
)	O
that	O
is	O
not	O
actually	O
prime	O
.	O
</s>
<s>
Pseudoprimes	B-Algorithm
are	O
classified	O
according	O
to	O
which	O
property	O
of	O
primes	O
they	O
satisfy	O
.	O
</s>
<s>
Some	O
sources	O
use	O
the	O
term	O
pseudoprime	B-Algorithm
to	O
describe	O
all	O
probable	B-Algorithm
primes	I-Algorithm
,	O
both	O
composite	O
numbers	O
and	O
actual	O
primes	O
.	O
</s>
<s>
Pseudoprimes	B-Algorithm
are	O
of	O
primary	O
importance	O
in	O
public-key	B-Application
cryptography	I-Application
,	O
which	O
makes	O
use	O
of	O
the	O
difficulty	O
of	O
factoring	O
large	O
numbers	O
into	O
their	O
prime	O
factors	O
.	O
</s>
<s>
But	O
finding	O
two	O
large	O
prime	O
numbers	O
as	O
needed	O
for	O
this	O
use	O
is	O
also	O
expensive	O
,	O
so	O
various	O
probabilistic	O
primality	B-Algorithm
tests	I-Algorithm
are	O
used	O
,	O
some	O
of	O
which	O
in	O
rare	O
cases	O
inappropriately	O
deliver	O
composite	O
numbers	O
instead	O
of	O
primes	O
.	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
deterministic	O
primality	B-Algorithm
tests	I-Algorithm
,	O
such	O
as	O
the	O
AKS	B-Algorithm
primality	I-Algorithm
test	I-Algorithm
,	O
do	O
not	O
give	O
false	O
positives	O
;	O
there	O
are	O
no	O
pseudoprimes	B-Algorithm
with	O
respect	O
to	O
them	O
.	O
</s>
<s>
Fermat	O
's	O
little	O
theorem	O
states	O
that	O
if	O
p	O
is	O
prime	O
and	O
a	O
is	O
coprime	O
to	O
p	O
,	O
then	O
ap−1−1	O
is	O
divisible	O
by	O
p	O
.	O
For	O
an	O
integer	O
a	O
>	O
1	O
,	O
if	O
a	O
composite	O
integer	O
x	O
divides	O
ax−1−1	O
,	O
then	O
x	O
is	O
called	O
a	O
Fermat	B-Algorithm
pseudoprime	I-Algorithm
to	O
base	O
a	O
.	O
</s>
<s>
It	O
follows	O
that	O
if	O
x	O
is	O
a	O
Fermat	B-Algorithm
pseudoprime	I-Algorithm
to	O
base	O
a	O
,	O
then	O
x	O
is	O
coprime	O
to	O
a	O
.	O
</s>
<s>
Some	O
sources	O
use	O
variations	O
of	O
this	O
definition	O
,	O
for	O
example	O
to	O
allow	O
only	O
odd	O
numbers	O
to	O
be	O
pseudoprimes	B-Algorithm
.	O
</s>
<s>
An	O
integer	O
x	O
that	O
is	O
a	O
Fermat	B-Algorithm
pseudoprime	I-Algorithm
to	O
all	O
values	O
of	O
a	O
that	O
are	O
coprime	O
to	O
x	O
is	O
called	O
a	O
Carmichael	O
number	O
.	O
</s>
