<s>
In	O
number	O
theory	O
,	O
a	O
probable	B-Algorithm
prime	I-Algorithm
(	O
PRP	O
)	O
is	O
an	O
integer	O
that	O
satisfies	O
a	O
specific	O
condition	O
that	O
is	O
satisfied	O
by	O
all	O
prime	O
numbers	O
,	O
but	O
which	O
is	O
not	O
satisfied	O
by	O
most	O
composite	O
numbers	O
.	O
</s>
<s>
Different	O
types	O
of	O
probable	B-Algorithm
primes	I-Algorithm
have	O
different	O
specific	O
conditions	O
.	O
</s>
<s>
While	O
there	O
may	O
be	O
probable	B-Algorithm
primes	I-Algorithm
that	O
are	O
composite	O
(	O
called	O
pseudoprimes	B-Algorithm
)	O
,	O
the	O
condition	O
is	O
generally	O
chosen	O
in	O
order	O
to	O
make	O
such	O
exceptions	O
rare	O
.	O
</s>
<s>
If	O
the	O
result	O
is	O
1	O
,	O
then	O
n	O
is	O
likely	O
to	O
be	O
prime	O
;	O
n	O
is	O
then	O
called	O
a	O
probable	B-Algorithm
prime	I-Algorithm
to	O
base	O
a	O
.	O
</s>
<s>
A	O
weak	O
probable	B-Algorithm
prime	I-Algorithm
to	O
base	O
a	O
is	O
an	O
integer	O
that	O
is	O
a	O
probable	B-Algorithm
prime	I-Algorithm
to	O
base	O
a	O
,	O
but	O
which	O
is	O
not	O
a	O
strong	O
probable	B-Algorithm
prime	I-Algorithm
to	O
base	O
a	O
(	O
see	O
below	O
)	O
.	O
</s>
<s>
For	O
a	O
fixed	O
base	O
a	O
,	O
it	O
is	O
unusual	O
for	O
a	O
composite	O
number	O
to	O
be	O
a	O
probable	B-Algorithm
prime	I-Algorithm
(	O
that	O
is	O
,	O
a	O
pseudoprime	B-Algorithm
)	O
to	O
that	O
base	O
.	O
</s>
<s>
For	O
example	O
,	O
up	O
to	O
,	O
there	O
are	O
11,408,012,595	O
odd	O
composite	O
numbers	O
,	O
but	O
only	O
21,853	O
pseudoprimes	B-Algorithm
base	O
2	O
.	O
</s>
<s>
Probable	O
primality	O
is	O
a	O
basis	O
for	O
efficient	O
primality	B-Algorithm
testing	I-Algorithm
algorithms	O
,	O
which	O
find	O
application	O
in	O
cryptography	O
.	O
</s>
<s>
These	O
algorithms	O
are	O
usually	O
probabilistic	B-General_Concept
in	O
nature	O
.	O
</s>
<s>
The	O
idea	O
is	O
that	O
while	O
there	O
are	O
composite	O
probable	B-Algorithm
primes	I-Algorithm
to	O
base	O
a	O
for	O
any	O
fixed	O
a	O
,	O
we	O
may	O
hope	O
there	O
exists	O
some	O
fixed	O
P1	O
such	O
that	O
for	O
any	O
given	O
composite	O
n	O
,	O
if	O
we	O
choose	O
a	O
at	O
random	O
,	O
then	O
the	O
probability	O
that	O
n	O
is	O
pseudoprime	B-Algorithm
to	O
base	O
a	O
is	O
at	O
most	O
P	O
.	O
If	O
we	O
repeat	O
this	O
test	O
k	O
times	O
,	O
choosing	O
a	O
new	O
a	O
each	O
time	O
,	O
the	O
probability	O
of	O
n	O
being	O
pseudoprime	B-Algorithm
to	O
all	O
the	O
as	O
tested	O
is	O
hence	O
at	O
most	O
Pk	O
,	O
and	O
as	O
this	O
decreases	O
exponentially	O
,	O
only	O
moderate	O
k	O
is	O
required	O
to	O
make	O
this	O
probability	O
negligibly	O
small	O
(	O
compared	O
to	O
,	O
for	O
example	O
,	O
the	O
probability	O
of	O
computer	O
hardware	O
error	O
)	O
.	O
</s>
<s>
Euler	O
probable	B-Algorithm
primes	I-Algorithm
(	O
P	O
=	O
1/2	O
,	O
Solovay	O
–	O
Strassen	O
algorithm	O
)	O
.	O
</s>
<s>
A	O
PRP	O
test	O
is	O
sometimes	O
combined	O
with	O
a	O
table	O
of	O
small	O
pseudoprimes	B-Algorithm
to	O
quickly	O
establish	O
the	O
primality	O
of	O
a	O
given	O
number	O
smaller	O
than	O
some	O
threshold	O
.	O
</s>
<s>
An	O
Euler	O
probable	B-Algorithm
prime	I-Algorithm
to	O
base	O
a	O
is	O
an	O
integer	O
that	O
is	O
indicated	O
prime	O
by	O
the	O
somewhat	O
stronger	O
theorem	O
that	O
for	O
any	O
prime	O
p	O
,	O
a( p−1	O
)	O
/2	O
equals	O
modulop	O
,	O
where	O
is	O
the	O
Jacobi	O
symbol	O
.	O
</s>
<s>
An	O
Euler	O
probable	B-Algorithm
prime	I-Algorithm
which	O
is	O
composite	O
is	O
called	O
an	O
Euler	O
–	O
Jacobi	O
pseudoprime	B-Algorithm
to	O
basea	O
.	O
</s>
<s>
The	O
smallest	O
Euler-Jacobi	O
pseudoprime	B-Algorithm
to	O
base	O
2	O
is	O
561	O
.	O
</s>
<s>
There	O
are	O
11347	O
Euler-Jacobi	O
pseudoprimes	B-Algorithm
base	O
2	O
that	O
are	O
less	O
than	O
25·109	O
.	O
</s>
<s>
The	O
number	O
n	O
is	O
a	O
strong	O
probable	B-Algorithm
prime	I-Algorithm
(	O
SPRP	O
)	O
to	O
base	O
a	O
if	O
:	O
</s>
<s>
A	O
composite	O
strong	O
probable	B-Algorithm
prime	I-Algorithm
to	O
base	O
a	O
is	O
called	O
a	O
strong	B-Algorithm
pseudoprime	I-Algorithm
to	O
base	O
a	O
.	O
</s>
<s>
Every	O
strong	O
probable	B-Algorithm
prime	I-Algorithm
to	O
base	O
a	O
is	O
also	O
an	O
Euler	O
probable	B-Algorithm
prime	I-Algorithm
to	O
the	O
same	O
base	O
,	O
but	O
not	O
vice	O
versa	O
.	O
</s>
<s>
The	O
smallest	O
strong	B-Algorithm
pseudoprime	I-Algorithm
base	O
2	O
is	O
2047	O
.	O
</s>
<s>
There	O
are	O
4842	O
strong	B-Algorithm
pseudoprimes	I-Algorithm
base	O
2	O
that	O
are	O
less	O
than	O
25·109	O
.	O
</s>
<s>
There	O
are	O
also	O
Lucas	O
probable	B-Algorithm
primes	I-Algorithm
,	O
which	O
are	O
based	O
on	O
Lucas	B-Algorithm
sequences	I-Algorithm
.	O
</s>
<s>
A	O
Lucas	B-Algorithm
probable	I-Algorithm
prime	I-Algorithm
test	O
can	O
be	O
used	O
alone	O
.	O
</s>
<s>
The	O
Baillie	B-Algorithm
–	I-Algorithm
PSW	I-Algorithm
primality	I-Algorithm
test	I-Algorithm
combines	O
a	O
Lucas	O
test	O
with	O
a	O
strong	O
probable	B-Algorithm
prime	I-Algorithm
test	O
.	O
</s>
<s>
To	O
test	O
whether	O
97	O
is	O
a	O
strong	O
probable	B-Algorithm
prime	I-Algorithm
base	O
2	O
:	O
</s>
<s>
Therefore	O
,	O
is	O
a	O
strong	O
probable	B-Algorithm
prime	I-Algorithm
base	O
2	O
(	O
and	O
is	O
therefore	O
a	O
probable	B-Algorithm
prime	I-Algorithm
base	O
2	O
)	O
.	O
</s>
