<s>
In	O
mathematics	O
,	O
Pépin	B-Algorithm
's	I-Algorithm
test	I-Algorithm
is	O
a	O
primality	B-Algorithm
test	I-Algorithm
,	O
which	O
can	O
be	O
used	O
to	O
determine	O
whether	O
a	O
Fermat	O
number	O
is	O
prime	O
.	O
</s>
<s>
It	O
is	O
a	O
variant	O
of	O
Proth	B-Algorithm
's	I-Algorithm
test	I-Algorithm
.	O
</s>
<s>
Pépin	B-Algorithm
's	I-Algorithm
test	I-Algorithm
states	O
that	O
for	O
n	O
>	O
0	O
,	O
</s>
<s>
The	O
expression	O
can	O
be	O
evaluated	O
modulo	O
by	O
repeated	B-Algorithm
squaring	I-Algorithm
.	O
</s>
<s>
In	O
fact	O
,	O
Pépin	B-Algorithm
's	I-Algorithm
test	I-Algorithm
is	O
the	O
same	O
as	O
the	O
Euler-Jacobi	B-Algorithm
test	I-Algorithm
for	O
Fermat	O
numbers	O
,	O
since	O
the	O
Jacobi	O
symbol	O
is	O
−1	O
,	O
i.e.	O
</s>
<s>
there	O
are	O
no	O
Fermat	O
numbers	O
which	O
are	O
Euler-Jacobi	B-Algorithm
pseudoprimes	I-Algorithm
to	O
these	O
bases	O
listed	O
above	O
.	O
</s>
<s>
By	O
repeated	B-Algorithm
squaring	I-Algorithm
,	O
we	O
find	O
that	O
,	O
thus	O
,	O
and	O
.	O
</s>
<s>
Because	O
of	O
the	O
sparsity	O
of	O
the	O
Fermat	O
numbers	O
,	O
the	O
Pépin	B-Algorithm
test	I-Algorithm
has	O
only	O
been	O
run	O
eight	O
times	O
(	O
on	O
Fermat	O
numbers	O
whose	O
primality	O
statuses	O
were	O
not	O
already	O
known	O
)	O
.	O
</s>
<s>
Mayer	O
,	O
Papadopoulos	O
and	O
Crandall	O
speculate	O
that	O
in	O
fact	O
,	O
because	O
of	O
the	O
size	O
of	O
the	O
still	O
undetermined	O
Fermat	O
numbers	O
,	O
it	O
will	O
take	O
considerable	O
advances	O
in	O
technology	O
before	O
any	O
more	O
Pépin	B-Algorithm
tests	I-Algorithm
can	O
be	O
run	O
in	O
a	O
reasonable	O
amount	O
of	O
time	O
.	O
</s>
<s>
Year	O
Provers	O
Fermat	O
number	O
Pépin	B-Algorithm
test	I-Algorithm
result	O
Factor	O
found	O
later	O
?	O
</s>
