<s>
In	O
number	O
theory	O
,	O
the	O
Fermat	B-Algorithm
pseudoprimes	I-Algorithm
make	O
up	O
the	O
most	O
important	O
class	O
of	O
pseudoprimes	B-Algorithm
that	O
come	O
from	O
Fermat	O
's	O
little	O
theorem	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>
In	O
other	O
words	O
,	O
a	O
composite	O
integer	O
is	O
a	O
Fermat	B-Algorithm
pseudoprime	I-Algorithm
to	O
base	O
a	O
if	O
it	O
successfully	O
passes	O
the	O
Fermat	B-Algorithm
primality	I-Algorithm
test	I-Algorithm
for	O
the	O
base	O
a	O
.	O
</s>
<s>
The	O
false	O
statement	O
that	O
all	O
numbers	O
that	O
pass	O
the	O
Fermat	B-Algorithm
primality	I-Algorithm
test	I-Algorithm
for	O
base	O
2	O
,	O
are	O
prime	O
,	O
is	O
called	O
the	O
Chinese	O
hypothesis	O
.	O
</s>
<s>
The	O
smallest	O
base-2	O
Fermat	B-Algorithm
pseudoprime	I-Algorithm
is	O
341	O
.	O
</s>
<s>
Fermat	B-Algorithm
primality	I-Algorithm
test	I-Algorithm
for	O
the	O
base	O
2	O
.	O
</s>
<s>
Pseudoprimes	B-Algorithm
to	O
base	O
2	O
are	O
sometimes	O
called	O
Sarrus	O
numbers	O
,	O
after	O
P	O
.	O
F	O
.	O
Sarrus	O
who	O
discovered	O
that	O
341	O
has	O
this	O
property	O
,	O
Poulet	B-Algorithm
numbers	I-Algorithm
,	O
after	O
P	O
.	O
Poulet	O
who	O
made	O
a	O
table	O
of	O
such	O
numbers	O
,	O
or	O
Fermatians	O
.	O
</s>
<s>
A	O
Fermat	B-Algorithm
pseudoprime	I-Algorithm
is	O
often	O
called	O
a	O
pseudoprime	B-Algorithm
,	O
with	O
the	O
modifier	O
Fermat	O
being	O
understood	O
.	O
</s>
<s>
An	O
integer	O
x	O
that	O
is	O
a	O
Fermat	B-Algorithm
pseudoprime	I-Algorithm
for	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>
<s>
There	O
are	O
infinitely	O
many	O
pseudoprimes	B-Algorithm
to	O
any	O
given	O
base	O
a	O
>	O
1	O
.	O
</s>
<s>
In	O
1904	O
,	O
Cipolla	O
showed	O
how	O
to	O
produce	O
an	O
infinite	O
number	O
of	O
pseudoprimes	B-Algorithm
base	O
a	O
>	O
1	O
:	O
Let	O
p	O
be	O
any	O
odd	O
prime	O
that	O
does	O
not	O
divide	O
a2	O
-	O
1	O
.	O
</s>
<s>
Then	O
n	O
=	O
AB	O
is	O
composite	O
,	O
and	O
is	O
a	O
pseudoprime	B-Algorithm
to	O
base	O
a	O
.	O
</s>
<s>
For	O
example	O
,	O
if	O
a	O
=	O
2	O
and	O
p	O
=	O
5	O
,	O
then	O
A	O
=	O
31	O
,	O
B	O
=	O
11	O
,	O
and	O
n	O
=	O
341	O
is	O
a	O
pseudoprime	B-Algorithm
to	O
base	O
2	O
.	O
</s>
<s>
There	O
are	O
three	O
pseudoprimes	B-Algorithm
to	O
base	O
2	O
below	O
1000	O
,	O
245	O
below	O
one	O
million	O
,	O
and	O
21853	O
less	O
than	O
25·109	O
.	O
</s>
<s>
There	O
are	O
4842	O
strong	B-Algorithm
pseudoprimes	I-Algorithm
base	O
2	O
and	O
2163	O
Carmichael	O
numbers	O
below	O
this	O
limit	O
(	O
see	O
Table	O
1	O
of	O
)	O
.	O
</s>
<s>
Starting	O
at	O
17·257	O
,	O
the	O
product	O
of	O
consecutive	O
Fermat	O
numbers	O
is	O
a	O
base-2	O
pseudoprime	B-Algorithm
,	O
and	O
so	O
are	O
all	O
Fermat	O
composites	O
and	O
Mersenne	O
composites	O
.	O
</s>
<s>
The	O
factorizations	O
of	O
the	O
60	O
Poulet	B-Algorithm
numbers	I-Algorithm
up	O
to	O
60787	O
,	O
including	O
13	O
Carmichael	O
numbers	O
(	O
in	O
bold	O
)	O
,	O
are	O
in	O
the	O
following	O
table	O
.	O
</s>
<s>
A	O
Poulet	B-Algorithm
number	I-Algorithm
all	O
of	O
whose	O
divisors	O
d	O
divide	O
2d	O
−	O
2	O
is	O
called	O
a	O
super-Poulet	O
number	O
.	O
</s>
<s>
There	O
are	O
infinitely	O
many	O
Poulet	B-Algorithm
numbers	I-Algorithm
which	O
are	O
not	O
super-Poulet	O
Numbers	O
.	O
</s>
<s>
The	O
smallest	O
pseudoprime	B-Algorithm
for	O
each	O
base	O
a	O
≤	O
200	O
is	O
given	O
in	O
the	O
following	O
table	O
;	O
the	O
colors	O
mark	O
the	O
number	O
of	O
prime	O
factors	O
.	O
</s>
<s>
Unlike	O
in	O
the	O
definition	O
at	O
the	O
start	O
of	O
the	O
article	O
,	O
pseudoprimes	B-Algorithm
below	O
a	O
are	O
excluded	O
in	O
the	O
table	O
.	O
</s>
<s>
nFirst	O
few	O
Fermat	B-Algorithm
pseudoprimes	I-Algorithm
in	O
base	O
nOEIS	O
sequence14	O
,	O
6	O
,	O
8	O
,	O
9	O
,	O
10	O
,	O
12	O
,	O
14	O
,	O
15	O
,	O
16	O
,	O
18	O
,	O
20	O
,	O
21	O
,	O
22	O
,	O
24	O
,	O
25	O
,	O
26	O
,	O
27	O
,	O
28	O
,	O
30	O
,	O
32	O
,	O
33	O
,	O
34	O
,	O
35	O
,	O
36	O
,	O
38	O
,	O
39	O
,	O
40	O
,	O
42	O
,	O
44	O
,	O
45	O
,	O
46	O
,	O
48	O
,	O
49	O
,	O
50	O
,	O
51	O
,	O
52	O
,	O
54	O
,	O
55	O
,	O
56	O
,	O
57	O
,	O
58	O
,	O
60	O
,	O
62	O
,	O
63	O
,	O
64	O
,	O
65	O
,	O
66	O
,	O
68	O
,	O
69	O
,	O
70	O
,	O
72	O
,	O
74	O
,	O
75	O
,	O
76	O
,	O
77	O
,	O
78	O
,	O
80	O
,	O
81	O
,	O
82	O
,	O
84	O
,	O
85	O
,	O
86	O
,	O
87	O
,	O
88	O
,	O
90	O
,	O
91	O
,	O
92	O
,	O
93	O
,	O
94	O
,	O
95	O
,	O
96	O
,	O
98	O
,	O
99	O
,	O
100	O
,	O
...	O
(	O
All	O
composites	O
)	O
2341	O
,	O
561	O
,	O
645	O
,	O
1105	O
,	O
1387	O
,	O
1729	O
,	O
1905	O
,	O
2047	O
,	O
2465	O
,	O
2701	O
,	O
2821	O
,	O
3277	O
,	O
4033	O
,	O
4369	O
,	O
4371	O
,	O
4681	O
,	O
5461	O
,	O
6601	O
,	O
7957	O
,	O
8321	O
,	O
8481	O
,	O
8911	O
,	O
391	O
...	O
,	O
121	O
,	O
286	O
,	O
671	O
,	O
703	O
,	O
949	O
,	O
1105	O
,	O
1541	O
,	O
1729	O
,	O
1891	O
,	O
2465	O
,	O
2665	O
,	O
2701	O
,	O
2821	O
,	O
3281	O
,	O
3367	O
,	O
3751	O
,	O
4961	O
,	O
5551	O
,	O
6601	O
,	O
7381	O
,	O
8401	O
,	O
8911	O
,	O
415	O
...	O
,	O
85	O
,	O
91	O
,	O
341	O
,	O
435	O
,	O
451	O
,	O
561	O
,	O
645	O
,	O
703	O
,	O
1105	O
,	O
1247	O
,	O
1271	O
,	O
1387	O
,	O
1581	O
,	O
1695	O
,	O
1729	O
,	O
1891	O
,	O
1905	O
,	O
2047	O
,	O
2071	O
,	O
2465	O
,	O
2701	O
,	O
2821	O
,	O
3133	O
,	O
3277	O
,	O
3367	O
,	O
3683	O
,	O
4033	O
,	O
4369	O
,	O
4371	O
,	O
4681	O
,	O
4795	O
,	O
4859	O
,	O
5461	O
,	O
5551	O
,	O
6601	O
,	O
6643	O
,	O
7957	O
,	O
8321	O
,	O
8481	O
,	O
8695	O
,	O
8911	O
,	O
9061	O
,	O
9131	O
,	O
9211	O
,	O
9605	O
,	O
9919	O
,	O
54	O
...	O
,	O
124	O
,	O
217	O
,	O
561	O
,	O
781	O
,	O
1541	O
,	O
1729	O
,	O
1891	O
,	O
2821	O
,	O
4123	O
,	O
5461	O
,	O
5611	O
,	O
5662	O
,	O
5731	O
,	O
6601	O
,	O
7449	O
,	O
7813	O
,	O
8029	O
,	O
8911	O
,	O
9881	O
,	O
635	O
...	O
,	O
185	O
,	O
217	O
,	O
301	O
,	O
481	O
,	O
1105	O
,	O
1111	O
,	O
1261	O
,	O
1333	O
,	O
1729	O
,	O
2465	O
,	O
2701	O
,	O
2821	O
,	O
3421	O
,	O
3565	O
,	O
3589	O
,	O
3913	O
,	O
4123	O
,	O
4495	O
,	O
5713	O
,	O
6533	O
,	O
6601	O
,	O
8029	O
,	O
8365	O
,	O
8911	O
,	O
9331	O
,	O
9881	O
,	O
76	O
...	O
,	O
25	O
,	O
325	O
,	O
561	O
,	O
703	O
,	O
817	O
,	O
1105	O
,	O
1825	O
,	O
2101	O
,	O
2353	O
,	O
2465	O
,	O
3277	O
,	O
4525	O
,	O
4825	O
,	O
6697	O
,	O
8321	O
,	O
89	O
...	O
,	O
21	O
,	O
45	O
,	O
63	O
,	O
65	O
,	O
105	O
,	O
117	O
,	O
133	O
,	O
153	O
,	O
231	O
,	O
273	O
,	O
341	O
,	O
481	O
,	O
511	O
,	O
561	O
,	O
585	O
,	O
645	O
,	O
651	O
,	O
861	O
,	O
949	O
,	O
1001	O
,	O
1105	O
,	O
1281	O
,	O
1365	O
,	O
1387	O
,	O
1417	O
,	O
1541	O
,	O
1649	O
,	O
1661	O
,	O
1729	O
,	O
1785	O
,	O
1905	O
,	O
2047	O
,	O
2169	O
,	O
2465	O
,	O
2501	O
,	O
2701	O
,	O
2821	O
,	O
3145	O
,	O
3171	O
,	O
3201	O
,	O
3277	O
,	O
3605	O
,	O
3641	O
,	O
4005	O
,	O
4033	O
,	O
4097	O
,	O
4369	O
,	O
4371	O
,	O
4641	O
,	O
4681	O
,	O
4921	O
,	O
5461	O
,	O
5565	O
,	O
5963	O
,	O
6305	O
,	O
6533	O
,	O
6601	O
,	O
6951	O
,	O
7107	O
,	O
7161	O
,	O
7957	O
,	O
8321	O
,	O
8481	O
,	O
8911	O
,	O
9265	O
,	O
9709	O
,	O
9773	O
,	O
9881	O
,	O
9945	O
,	O
94	O
...	O
,	O
8	O
,	O
28	O
,	O
52	O
,	O
91	O
,	O
121	O
,	O
205	O
,	O
286	O
,	O
364	O
,	O
511	O
,	O
532	O
,	O
616	O
,	O
671	O
,	O
697	O
,	O
703	O
,	O
946	O
,	O
949	O
,	O
1036	O
,	O
1105	O
,	O
1288	O
,	O
1387	O
,	O
1541	O
,	O
1729	O
,	O
1891	O
,	O
2465	O
,	O
2501	O
,	O
2665	O
,	O
2701	O
,	O
2806	O
,	O
2821	O
,	O
2926	O
,	O
3052	O
,	O
3281	O
,	O
3367	O
,	O
3751	O
,	O
4376	O
,	O
4636	O
,	O
4961	O
,	O
5356	O
,	O
5551	O
,	O
6364	O
,	O
6601	O
,	O
6643	O
,	O
7081	O
,	O
7381	O
,	O
7913	O
,	O
8401	O
,	O
8695	O
,	O
8744	O
,	O
8866	O
,	O
8911	O
,	O
109	O
...	O
,	O
33	O
,	O
91	O
,	O
99	O
,	O
259	O
,	O
451	O
,	O
481	O
,	O
561	O
,	O
657	O
,	O
703	O
,	O
909	O
,	O
1233	O
,	O
1729	O
,	O
2409	O
,	O
2821	O
,	O
2981	O
,	O
3333	O
,	O
3367	O
,	O
4141	O
,	O
4187	O
,	O
4521	O
,	O
5461	O
,	O
6533	O
,	O
6541	O
,	O
6601	O
,	O
7107	O
,	O
7471	O
,	O
7777	O
,	O
8149	O
,	O
8401	O
,	O
8911	O
,	O
1110	O
...	O
,	O
15	O
,	O
70	O
,	O
133	O
,	O
190	O
,	O
259	O
,	O
305	O
,	O
481	O
,	O
645	O
,	O
703	O
,	O
793	O
,	O
1105	O
,	O
1330	O
,	O
1729	O
,	O
2047	O
,	O
2257	O
,	O
2465	O
,	O
2821	O
,	O
4577	O
,	O
4921	O
,	O
5041	O
,	O
5185	O
,	O
6601	O
,	O
7869	O
,	O
8113	O
,	O
8170	O
,	O
8695	O
,	O
8911	O
,	O
9730	O
,	O
1265	O
...	O
,	O
91	O
,	O
133	O
,	O
143	O
,	O
145	O
,	O
247	O
,	O
377	O
,	O
385	O
,	O
703	O
,	O
1045	O
,	O
1099	O
,	O
1105	O
,	O
1649	O
,	O
1729	O
,	O
1885	O
,	O
1891	O
,	O
2041	O
,	O
2233	O
,	O
2465	O
,	O
2701	O
,	O
2821	O
,	O
2983	O
,	O
3367	O
,	O
3553	O
,	O
5005	O
,	O
5365	O
,	O
5551	O
,	O
5785	O
,	O
6061	O
,	O
6305	O
,	O
6601	O
,	O
8911	O
,	O
9073	O
,	O
134	O
...	O
,	O
6	O
,	O
12	O
,	O
21	O
,	O
85	O
,	O
105	O
,	O
231	O
,	O
244	O
,	O
276	O
,	O
357	O
,	O
427	O
,	O
561	O
,	O
1099	O
,	O
1785	O
,	O
1891	O
,	O
2465	O
,	O
2806	O
,	O
3605	O
,	O
5028	O
,	O
5149	O
,	O
5185	O
,	O
5565	O
,	O
6601	O
,	O
7107	O
,	O
8841	O
,	O
8911	O
,	O
9577	O
,	O
9637	O
,	O
1415	O
...	O
,	O
39	O
,	O
65	O
,	O
195	O
,	O
481	O
,	O
561	O
,	O
781	O
,	O
793	O
,	O
841	O
,	O
985	O
,	O
1105	O
,	O
1111	O
,	O
1541	O
,	O
1891	O
,	O
2257	O
,	O
2465	O
,	O
2561	O
,	O
2665	O
,	O
2743	O
,	O
3277	O
,	O
5185	O
,	O
5713	O
,	O
6501	O
,	O
6533	O
,	O
6541	O
,	O
7107	O
,	O
7171	O
,	O
7449	O
,	O
7543	O
,	O
7585	O
,	O
8321	O
,	O
9073	O
,	O
1514	O
...	O
,	O
341	O
,	O
742	O
,	O
946	O
,	O
1477	O
,	O
1541	O
,	O
1687	O
,	O
1729	O
,	O
1891	O
,	O
1921	O
,	O
2821	O
,	O
3133	O
,	O
3277	O
,	O
4187	O
,	O
6541	O
,	O
6601	O
,	O
7471	O
,	O
8701	O
,	O
8911	O
,	O
9073	O
,	O
1615	O
...	O
,	O
51	O
,	O
85	O
,	O
91	O
,	O
255	O
,	O
341	O
,	O
435	O
,	O
451	O
,	O
561	O
,	O
595	O
,	O
645	O
,	O
703	O
,	O
1105	O
,	O
1247	O
,	O
1261	O
,	O
1271	O
,	O
1285	O
,	O
1387	O
,	O
1581	O
,	O
1687	O
,	O
1695	O
,	O
1729	O
,	O
1891	O
,	O
1905	O
,	O
2047	O
,	O
2071	O
,	O
2091	O
,	O
2431	O
,	O
2465	O
,	O
2701	O
,	O
2821	O
,	O
3133	O
,	O
3277	O
,	O
3367	O
,	O
3655	O
,	O
3683	O
,	O
4033	O
,	O
4369	O
,	O
4371	O
,	O
4681	O
,	O
4795	O
,	O
4859	O
,	O
5083	O
,	O
5151	O
,	O
5461	O
,	O
5551	O
,	O
6601	O
,	O
6643	O
,	O
7471	O
,	O
7735	O
,	O
7957	O
,	O
8119	O
,	O
8227	O
,	O
8245	O
,	O
8321	O
,	O
8481	O
,	O
8695	O
,	O
8749	O
,	O
8911	O
,	O
9061	O
,	O
9131	O
,	O
9211	O
,	O
9605	O
,	O
9919	O
,	O
174	O
...	O
,	O
8	O
,	O
9	O
,	O
16	O
,	O
45	O
,	O
91	O
,	O
145	O
,	O
261	O
,	O
781	O
,	O
1111	O
,	O
1228	O
,	O
1305	O
,	O
1729	O
,	O
1885	O
,	O
2149	O
,	O
2821	O
,	O
3991	O
,	O
4005	O
,	O
4033	O
,	O
4187	O
,	O
4912	O
,	O
5365	O
,	O
5662	O
,	O
5833	O
,	O
6601	O
,	O
6697	O
,	O
7171	O
,	O
8481	O
,	O
8911	O
,	O
1825	O
...	O
,	O
49	O
,	O
65	O
,	O
85	O
,	O
133	O
,	O
221	O
,	O
323	O
,	O
325	O
,	O
343	O
,	O
425	O
,	O
451	O
,	O
637	O
,	O
931	O
,	O
1105	O
,	O
1225	O
,	O
1369	O
,	O
1387	O
,	O
1649	O
,	O
1729	O
,	O
1921	O
,	O
2149	O
,	O
2465	O
,	O
2701	O
,	O
2821	O
,	O
2825	O
,	O
2977	O
,	O
3325	O
,	O
4165	O
,	O
4577	O
,	O
4753	O
,	O
5525	O
,	O
5725	O
,	O
5833	O
,	O
5941	O
,	O
6305	O
,	O
6517	O
,	O
6601	O
,	O
7345	O
,	O
8911	O
,	O
9061	O
,	O
196	O
...	O
,	O
9	O
,	O
15	O
,	O
18	O
,	O
45	O
,	O
49	O
,	O
153	O
,	O
169	O
,	O
343	O
,	O
561	O
,	O
637	O
,	O
889	O
,	O
905	O
,	O
906	O
,	O
1035	O
,	O
1105	O
,	O
1629	O
,	O
1661	O
,	O
1849	O
,	O
1891	O
,	O
2353	O
,	O
2465	O
,	O
2701	O
,	O
2821	O
,	O
2955	O
,	O
3201	O
,	O
4033	O
,	O
4681	O
,	O
5461	O
,	O
5466	O
,	O
5713	O
,	O
6223	O
,	O
6541	O
,	O
6601	O
,	O
6697	O
,	O
7957	O
,	O
8145	O
,	O
8281	O
,	O
8401	O
,	O
8869	O
,	O
9211	O
,	O
9997	O
,	O
2021	O
...	O
,	O
57	O
,	O
133	O
,	O
231	O
,	O
399	O
,	O
561	O
,	O
671	O
,	O
861	O
,	O
889	O
,	O
1281	O
,	O
1653	O
,	O
1729	O
,	O
1891	O
,	O
2059	O
,	O
2413	O
,	O
2501	O
,	O
2761	O
,	O
2821	O
,	O
2947	O
,	O
3059	O
,	O
3201	O
,	O
4047	O
,	O
5271	O
,	O
5461	O
,	O
5473	O
,	O
5713	O
,	O
5833	O
,	O
6601	O
,	O
6817	O
,	O
7999	O
,	O
8421	O
,	O
8911	O
,	O
214	O
...	O
,	O
10	O
,	O
20	O
,	O
55	O
,	O
65	O
,	O
85	O
,	O
221	O
,	O
703	O
,	O
793	O
,	O
1045	O
,	O
1105	O
,	O
1852	O
,	O
2035	O
,	O
2465	O
,	O
3781	O
,	O
4630	O
,	O
5185	O
,	O
5473	O
,	O
5995	O
,	O
6541	O
,	O
7363	O
,	O
8695	O
,	O
8965	O
,	O
9061	O
,	O
2221	O
...	O
,	O
69	O
,	O
91	O
,	O
105	O
,	O
161	O
,	O
169	O
,	O
345	O
,	O
483	O
,	O
485	O
,	O
645	O
,	O
805	O
,	O
1105	O
,	O
1183	O
,	O
1247	O
,	O
1261	O
,	O
1541	O
,	O
1649	O
,	O
1729	O
,	O
1891	O
,	O
2037	O
,	O
2041	O
,	O
2047	O
,	O
2413	O
,	O
2465	O
,	O
2737	O
,	O
2821	O
,	O
3241	O
,	O
3605	O
,	O
3801	O
,	O
5551	O
,	O
5565	O
,	O
5963	O
,	O
6019	O
,	O
6601	O
,	O
6693	O
,	O
7081	O
,	O
7107	O
,	O
7267	O
,	O
7665	O
,	O
8119	O
,	O
8365	O
,	O
8421	O
,	O
8911	O
,	O
9453	O
,	O
2322	O
...	O
,	O
33	O
,	O
91	O
,	O
154	O
,	O
165	O
,	O
169	O
,	O
265	O
,	O
341	O
,	O
385	O
,	O
451	O
,	O
481	O
,	O
553	O
,	O
561	O
,	O
638	O
,	O
946	O
,	O
1027	O
,	O
1045	O
,	O
1065	O
,	O
1105	O
,	O
1183	O
,	O
1271	O
,	O
1729	O
,	O
1738	O
,	O
1749	O
,	O
2059	O
,	O
2321	O
,	O
2465	O
,	O
2501	O
,	O
2701	O
,	O
2821	O
,	O
2926	O
,	O
3097	O
,	O
3445	O
,	O
4033	O
,	O
4081	O
,	O
4345	O
,	O
4371	O
,	O
4681	O
,	O
5005	O
,	O
5149	O
,	O
6253	O
,	O
6369	O
,	O
6533	O
,	O
6541	O
,	O
7189	O
,	O
7267	O
,	O
7957	O
,	O
8321	O
,	O
8365	O
,	O
8651	O
,	O
8745	O
,	O
8911	O
,	O
8965	O
,	O
9805	O
,	O
2425	O
...	O
,	O
115	O
,	O
175	O
,	O
325	O
,	O
553	O
,	O
575	O
,	O
805	O
,	O
949	O
,	O
1105	O
,	O
1541	O
,	O
1729	O
,	O
1771	O
,	O
1825	O
,	O
1975	O
,	O
2413	O
,	O
2425	O
,	O
2465	O
,	O
2701	O
,	O
2737	O
,	O
2821	O
,	O
2885	O
,	O
3781	O
,	O
4207	O
,	O
4537	O
,	O
6601	O
,	O
6931	O
,	O
6943	O
,	O
7081	O
,	O
7189	O
,	O
7471	O
,	O
7501	O
,	O
7813	O
,	O
8725	O
,	O
8911	O
,	O
9085	O
,	O
9361	O
,	O
9809	O
,	O
254	O
...	O
,	O
6	O
,	O
8	O
,	O
12	O
,	O
24	O
,	O
28	O
,	O
39	O
,	O
66	O
,	O
91	O
,	O
124	O
,	O
217	O
,	O
232	O
,	O
276	O
,	O
403	O
,	O
426	O
,	O
451	O
,	O
532	O
,	O
561	O
,	O
616	O
,	O
703	O
,	O
781	O
,	O
804	O
,	O
868	O
,	O
946	O
,	O
1128	O
,	O
1288	O
,	O
1541	O
,	O
1729	O
,	O
1891	O
,	O
2047	O
,	O
2701	O
,	O
2806	O
,	O
2821	O
,	O
2911	O
,	O
2926	O
,	O
3052	O
,	O
3126	O
,	O
3367	O
,	O
3592	O
,	O
3976	O
,	O
4069	O
,	O
4123	O
,	O
4207	O
,	O
4564	O
,	O
4636	O
,	O
4686	O
,	O
5321	O
,	O
5461	O
,	O
5551	O
,	O
5611	O
,	O
5662	O
,	O
5731	O
,	O
5963	O
,	O
6601	O
,	O
7449	O
,	O
7588	O
,	O
7813	O
,	O
8029	O
,	O
8646	O
,	O
8911	O
,	O
9881	O
,	O
9976	O
,	O
269	O
...	O
,	O
15	O
,	O
25	O
,	O
27	O
,	O
45	O
,	O
75	O
,	O
133	O
,	O
135	O
,	O
153	O
,	O
175	O
,	O
217	O
,	O
225	O
,	O
259	O
,	O
425	O
,	O
475	O
,	O
561	O
,	O
589	O
,	O
675	O
,	O
703	O
,	O
775	O
,	O
925	O
,	O
1035	O
,	O
1065	O
,	O
1147	O
,	O
2465	O
,	O
3145	O
,	O
3325	O
,	O
3385	O
,	O
3565	O
,	O
3825	O
,	O
4123	O
,	O
4525	O
,	O
4741	O
,	O
4921	O
,	O
5041	O
,	O
5425	O
,	O
6093	O
,	O
6475	O
,	O
6525	O
,	O
6601	O
,	O
6697	O
,	O
8029	O
,	O
8695	O
,	O
8911	O
,	O
9073	O
,	O
2726	O
...	O
,	O
65	O
,	O
91	O
,	O
121	O
,	O
133	O
,	O
247	O
,	O
259	O
,	O
286	O
,	O
341	O
,	O
365	O
,	O
481	O
,	O
671	O
,	O
703	O
,	O
949	O
,	O
1001	O
,	O
1105	O
,	O
1541	O
,	O
1649	O
,	O
1729	O
,	O
1891	O
,	O
2071	O
,	O
2465	O
,	O
2665	O
,	O
2701	O
,	O
2821	O
,	O
2981	O
,	O
2993	O
,	O
3146	O
,	O
3281	O
,	O
3367	O
,	O
3605	O
,	O
3751	O
,	O
4033	O
,	O
4745	O
,	O
4921	O
,	O
4961	O
,	O
5299	O
,	O
5461	O
,	O
5551	O
,	O
5611	O
,	O
5621	O
,	O
6305	O
,	O
6533	O
,	O
6601	O
,	O
7381	O
,	O
7585	O
,	O
7957	O
,	O
8227	O
,	O
8321	O
,	O
8401	O
,	O
8911	O
,	O
9139	O
,	O
9709	O
,	O
9809	O
,	O
9841	O
,	O
9881	O
,	O
9919	O
,	O
289	O
...	O
,	O
27	O
,	O
45	O
,	O
87	O
,	O
145	O
,	O
261	O
,	O
361	O
,	O
529	O
,	O
561	O
,	O
703	O
,	O
783	O
,	O
785	O
,	O
1105	O
,	O
1305	O
,	O
1413	O
,	O
1431	O
,	O
1885	O
,	O
2041	O
,	O
2413	O
,	O
2465	O
,	O
2871	O
,	O
3201	O
,	O
3277	O
,	O
4553	O
,	O
4699	O
,	O
5149	O
,	O
5181	O
,	O
5365	O
,	O
7065	O
,	O
8149	O
,	O
8321	O
,	O
8401	O
,	O
9841	O
,	O
294	O
...	O
,	O
14	O
,	O
15	O
,	O
21	O
,	O
28	O
,	O
35	O
,	O
52	O
,	O
91	O
,	O
105	O
,	O
231	O
,	O
268	O
,	O
341	O
,	O
364	O
,	O
469	O
,	O
481	O
,	O
561	O
,	O
651	O
,	O
793	O
,	O
871	O
,	O
1105	O
,	O
1729	O
,	O
1876	O
,	O
1897	O
,	O
2105	O
,	O
2257	O
,	O
2821	O
,	O
3484	O
,	O
3523	O
,	O
4069	O
,	O
4371	O
,	O
4411	O
,	O
5149	O
,	O
5185	O
,	O
5356	O
,	O
5473	O
,	O
5565	O
,	O
5611	O
,	O
6097	O
,	O
6601	O
,	O
7161	O
,	O
7294	O
,	O
8321	O
,	O
8401	O
,	O
8421	O
,	O
8841	O
,	O
8911	O
,	O
3049	O
...	O
,	O
91	O
,	O
133	O
,	O
217	O
,	O
247	O
,	O
341	O
,	O
403	O
,	O
469	O
,	O
493	O
,	O
589	O
,	O
637	O
,	O
703	O
,	O
871	O
,	O
899	O
,	O
901	O
,	O
931	O
,	O
1273	O
,	O
1519	O
,	O
1537	O
,	O
1729	O
,	O
2059	O
,	O
2077	O
,	O
2821	O
,	O
3097	O
,	O
3277	O
,	O
3283	O
,	O
3367	O
,	O
3577	O
,	O
4081	O
,	O
4097	O
,	O
4123	O
,	O
5729	O
,	O
6031	O
,	O
6061	O
,	O
6097	O
,	O
6409	O
,	O
6601	O
,	O
6817	O
,	O
7657	O
,	O
8023	O
,	O
8029	O
,	O
8401	O
,	O
8911	O
,	O
9881	O
,	O
...	O
</s>
<s>
If	O
composite	O
is	O
even	O
,	O
then	O
is	O
a	O
Fermat	B-Algorithm
pseudoprime	I-Algorithm
to	O
the	O
trivial	O
base	O
.	O
</s>
<s>
If	O
composite	O
is	O
odd	O
,	O
then	O
is	O
a	O
Fermat	B-Algorithm
pseudoprime	I-Algorithm
to	O
the	O
trivial	O
bases	O
.	O
</s>
<s>
Every	O
odd	O
composite	O
is	O
a	O
Fermat	B-Algorithm
pseudoprime	I-Algorithm
to	O
at	O
least	O
two	O
nontrivial	O
bases	O
modulo	O
unless	O
is	O
a	O
power	O
of	O
3	O
.	O
</s>
<s>
For	O
composite	O
n	O
<	O
200	O
,	O
the	O
following	O
is	O
a	O
table	O
of	O
all	O
bases	O
b	O
<	O
n	O
which	O
n	O
is	O
a	O
Fermat	B-Algorithm
pseudoprime	I-Algorithm
.	O
</s>
<s>
If	O
a	O
composite	O
number	O
n	O
is	O
not	O
in	O
the	O
table	O
(	O
or	O
n	O
is	O
in	O
the	O
sequence	O
)	O
,	O
then	O
n	O
is	O
a	O
pseudoprime	B-Algorithm
only	O
to	O
the	O
trivial	O
base	O
1	O
modulo	O
n	O
.	O
</s>
<s>
For	O
more	O
information	O
(	O
n	O
=	O
201	O
to	O
5000	O
)	O
,	O
see	O
,	O
this	O
page	O
does	O
not	O
define	O
n	O
is	O
a	O
pseudoprime	B-Algorithm
to	O
a	O
base	O
congruent	O
to	O
1	O
or	O
-1	O
(	O
mod	O
n	O
)	O
.	O
</s>
<s>
When	O
p	O
is	O
a	O
prime	O
,	O
p2	O
is	O
a	O
Fermat	B-Algorithm
pseudoprime	I-Algorithm
to	O
base	O
b	O
if	O
and	O
only	O
if	O
p	O
is	O
a	O
Wieferich	O
prime	O
to	O
base	O
b	O
.	O
</s>
<s>
For	O
example	O
,	O
10932	O
=	O
1194649	O
is	O
a	O
Fermat	B-Algorithm
pseudoprime	I-Algorithm
to	O
base	O
2	O
,	O
and	O
112	O
=	O
121	O
is	O
a	O
Fermat	B-Algorithm
pseudoprime	I-Algorithm
to	O
base	O
3	O
.	O
</s>
<s>
A	O
composite	O
number	O
n	O
which	O
satisfy	O
that	O
is	O
called	O
weak	O
pseudoprime	B-Algorithm
to	O
base	O
b	O
.	O
</s>
<s>
A	O
pseudoprime	B-Algorithm
to	O
base	O
a	O
(	O
under	O
the	O
usual	O
definition	O
)	O
satisfies	O
this	O
condition	O
.	O
</s>
<s>
Conversely	O
,	O
a	O
weak	O
pseudoprime	B-Algorithm
that	O
is	O
coprime	O
with	O
the	O
base	O
is	O
a	O
pseudoprime	B-Algorithm
in	O
the	O
usual	O
sense	O
,	O
otherwise	O
this	O
may	O
or	O
may	O
not	O
be	O
the	O
case	O
.	O
</s>
<s>
The	O
least	O
weak	O
pseudoprime	B-Algorithm
to	O
base	O
b	O
=	O
1	O
,	O
2	O
,	O
...	O
are	O
:	O
</s>
<s>
(	O
see	O
)	O
Besides	O
,	O
the	O
smallest	O
pseudoprime	B-Algorithm
to	O
base	O
n	O
(	O
also	O
not	O
necessary	O
exceeding	O
n	O
)	O
(	O
)	O
is	O
also	O
usually	O
semiprime	O
,	O
the	O
first	O
counterexample	O
is	O
(	O
648	O
)	O
=	O
385	O
=	O
5	O
×	O
7	O
×	O
11	O
.	O
</s>
<s>
Carmichael	O
numbers	O
are	O
weak	O
pseudoprimes	B-Algorithm
to	O
all	O
bases	O
.	O
</s>
<s>
The	O
smallest	O
even	O
weak	O
pseudoprime	B-Algorithm
in	O
base	O
2	O
is	O
161038	O
(	O
see	O
)	O
.	O
</s>
<s>
Another	O
approach	O
is	O
to	O
use	O
more	O
refined	O
notions	O
of	O
pseudoprimality	B-Algorithm
,	O
e.g.	O
</s>
<s>
strong	B-Algorithm
pseudoprimes	I-Algorithm
or	O
Euler	B-Algorithm
–	I-Algorithm
Jacobi	I-Algorithm
pseudoprimes	I-Algorithm
,	O
for	O
which	O
there	O
are	O
no	O
analogues	O
of	O
Carmichael	O
numbers	O
.	O
</s>
<s>
This	O
leads	O
to	O
probabilistic	B-General_Concept
algorithms	I-General_Concept
such	O
as	O
the	O
Solovay	O
–	O
Strassen	O
primality	B-Algorithm
test	I-Algorithm
,	O
the	O
Baillie	B-Algorithm
–	I-Algorithm
PSW	I-Algorithm
primality	I-Algorithm
test	I-Algorithm
,	O
and	O
the	O
Miller	B-Algorithm
–	I-Algorithm
Rabin	I-Algorithm
primality	I-Algorithm
test	I-Algorithm
,	O
which	O
produce	O
what	O
are	O
known	O
as	O
industrial-grade	O
primes	O
.	O
</s>
<s>
rigorously	O
proven	O
)	O
,	O
but	O
have	O
undergone	O
a	O
test	B-Algorithm
such	O
as	O
the	O
Miller	B-Algorithm
–	I-Algorithm
Rabin	I-Algorithm
test	I-Algorithm
which	O
has	O
nonzero	O
,	O
but	O
arbitrarily	O
low	O
,	O
probability	O
of	O
failure	O
.	O
</s>
<s>
The	O
rarity	O
of	O
such	O
pseudoprimes	B-Algorithm
has	O
important	O
practical	O
implications	O
.	O
</s>
<s>
For	O
example	O
,	O
public-key	B-Application
cryptography	I-Application
algorithms	O
such	O
as	O
RSA	B-Architecture
require	O
the	O
ability	O
to	O
quickly	O
find	O
large	O
primes	O
.	O
</s>
<s>
The	O
usual	O
algorithm	O
to	O
generate	O
prime	O
numbers	O
is	O
to	O
generate	O
random	O
odd	O
numbers	O
and	O
test	B-Algorithm
them	O
for	O
primality	O
.	O
</s>
<s>
However	O
,	O
deterministic	B-General_Concept
primality	B-Algorithm
tests	I-Algorithm
are	O
slow	O
.	O
</s>
<s>
If	O
the	O
user	O
is	O
willing	O
to	O
tolerate	O
an	O
arbitrarily	O
small	O
chance	O
that	O
the	O
number	O
found	O
is	O
not	O
a	O
prime	O
number	O
but	O
a	O
pseudoprime	B-Algorithm
,	O
it	O
is	O
possible	O
to	O
use	O
the	O
much	O
faster	O
and	O
simpler	O
Fermat	B-Algorithm
primality	I-Algorithm
test	I-Algorithm
.	O
</s>
