<s>
The	O
equation	O
can	O
be	O
tested	O
rather	O
quickly	O
,	O
which	O
can	O
be	O
used	O
for	O
probabilistic	O
primality	B-Algorithm
testing	I-Algorithm
.	O
</s>
<s>
Every	O
Euler	B-Algorithm
pseudoprime	I-Algorithm
is	O
also	O
a	O
Fermat	B-Algorithm
pseudoprime	I-Algorithm
.	O
</s>
<s>
It	O
is	O
not	O
possible	O
to	O
produce	O
a	O
definite	O
test	O
of	O
primality	O
based	O
on	O
whether	O
a	O
number	O
is	O
an	O
Euler	B-Algorithm
pseudoprime	I-Algorithm
because	O
there	O
exist	O
absolute	O
Euler	B-Algorithm
pseudoprimes	I-Algorithm
,	O
numbers	O
which	O
are	O
Euler	B-Algorithm
pseudoprimes	I-Algorithm
to	O
every	O
base	O
relatively	O
prime	O
to	O
themselves	O
.	O
</s>
<s>
The	O
absolute	O
Euler	B-Algorithm
pseudoprimes	I-Algorithm
are	O
a	O
subset	O
of	O
the	O
absolute	O
Fermat	B-Algorithm
pseudoprimes	I-Algorithm
,	O
or	O
Carmichael	O
numbers	O
,	O
and	O
the	O
smallest	O
absolute	O
Euler	B-Algorithm
pseudoprime	I-Algorithm
is	O
1729	O
=	O
71319	O
.	O
</s>
<s>
where	O
n	O
is	O
an	O
odd	O
composite	O
,	O
the	O
greatest	O
common	O
divisor	O
of	O
a	O
and	O
n	O
equals	O
1	O
,	O
and	O
(	O
a/n	O
)	O
is	O
the	O
Jacobi	O
symbol	O
,	O
is	O
the	O
more	O
common	O
definition	O
of	O
an	O
Euler	B-Algorithm
pseudoprime	I-Algorithm
.	O
</s>
<s>
A	O
discussion	O
of	O
numbers	O
of	O
this	O
form	O
can	O
be	O
found	O
at	O
Euler	B-Algorithm
–	I-Algorithm
Jacobi	I-Algorithm
pseudoprime	I-Algorithm
.	O
</s>
<s>
There	O
are	O
no	O
absolute	O
Euler	B-Algorithm
–	I-Algorithm
Jacobi	I-Algorithm
pseudoprimes	I-Algorithm
.	O
</s>
<s>
A	O
strong	B-Algorithm
probable	I-Algorithm
prime	I-Algorithm
test	O
is	O
even	O
stronger	O
than	O
the	O
Euler-Jacobi	O
test	O
but	O
takes	O
the	O
same	O
computational	O
effort	O
.	O
</s>
<s>
nEuler	O
pseudoprimes	B-Algorithm
to	O
base	O
n19	O
,	O
15	O
,	O
21	O
,	O
25	O
,	O
27	O
,	O
33	O
,	O
35	O
,	O
39	O
,	O
45	O
,	O
49	O
,	O
51	O
,	O
55	O
,	O
57	O
,	O
63	O
,	O
65	O
,	O
69	O
,	O
75	O
,	O
77	O
,	O
81	O
,	O
85	O
,	O
87	O
,	O
91	O
,	O
93	O
,	O
95	O
,	O
99	O
,	O
105	O
,	O
111	O
,	O
115	O
,	O
117	O
,	O
119	O
,	O
121	O
,	O
123	O
,	O
125	O
,	O
129	O
,	O
133	O
,	O
135	O
,	O
141	O
,	O
143	O
,	O
145	O
,	O
147	O
,	O
153	O
,	O
155	O
,	O
159	O
,	O
161	O
,	O
165	O
,	O
169	O
,	O
171	O
,	O
175	O
,	O
177	O
,	O
183	O
,	O
185	O
,	O
187	O
,	O
189	O
,	O
195	O
,	O
201	O
,	O
203	O
,	O
205	O
,	O
207	O
,	O
209	O
,	O
213	O
,	O
215	O
,	O
217	O
,	O
219	O
,	O
221	O
,	O
225	O
,	O
231	O
,	O
235	O
,	O
237	O
,	O
243	O
,	O
245	O
,	O
247	O
,	O
249	O
,	O
253	O
,	O
255	O
,	O
259	O
,	O
261	O
,	O
265	O
,	O
267	O
,	O
273	O
,	O
275	O
,	O
279	O
,	O
285	O
,	O
287	O
,	O
289	O
,	O
291	O
,	O
295	O
,	O
297	O
,	O
299	O
,	O
...	O
(	O
all	O
odd	O
composites	O
)	O
2341	O
,	O
561	O
,	O
1105	O
,	O
1729	O
,	O
1905	O
,	O
2047	O
,	O
2465	O
,	O
3277	O
,	O
4033	O
,	O
4681	O
,	O
5461	O
,	O
6601	O
,	O
8321	O
,	O
8481	O
,	O
3121	O
...	O
,	O
703	O
,	O
1541	O
,	O
1729	O
,	O
1891	O
,	O
2465	O
,	O
2821	O
,	O
3281	O
,	O
4961	O
,	O
7381	O
,	O
8401	O
,	O
8911	O
,	O
4341	O
...	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
5217	O
...	O
,	O
781	O
,	O
1541	O
,	O
1729	O
,	O
5461	O
,	O
5611	O
,	O
6601	O
,	O
7449	O
,	O
7813	O
,	O
6185	O
...	O
,	O
217	O
,	O
301	O
,	O
481	O
,	O
1111	O
,	O
1261	O
,	O
1333	O
,	O
1729	O
,	O
2465	O
,	O
2701	O
,	O
3421	O
,	O
3565	O
,	O
3589	O
,	O
3913	O
,	O
5713	O
,	O
6533	O
,	O
8365	O
,	O
725	O
...	O
,	O
325	O
,	O
703	O
,	O
817	O
,	O
1825	O
,	O
2101	O
,	O
2353	O
,	O
2465	O
,	O
3277	O
,	O
4525	O
,	O
6697	O
,	O
8321	O
,	O
89	O
...	O
,	O
21	O
,	O
65	O
,	O
105	O
,	O
133	O
,	O
273	O
,	O
341	O
,	O
481	O
,	O
511	O
,	O
561	O
,	O
585	O
,	O
1001	O
,	O
1105	O
,	O
1281	O
,	O
1417	O
,	O
1541	O
,	O
1661	O
,	O
1729	O
,	O
1905	O
,	O
2047	O
,	O
2465	O
,	O
2501	O
,	O
3201	O
,	O
3277	O
,	O
3641	O
,	O
4033	O
,	O
4097	O
,	O
4641	O
,	O
4681	O
,	O
4921	O
,	O
5461	O
,	O
6305	O
,	O
6533	O
,	O
6601	O
,	O
7161	O
,	O
8321	O
,	O
8481	O
,	O
9265	O
,	O
9709	O
,	O
991	O
...	O
,	O
121	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
109	O
...	O
,	O
33	O
,	O
91	O
,	O
481	O
,	O
657	O
,	O
1233	O
,	O
1729	O
,	O
2821	O
,	O
2981	O
,	O
4187	O
,	O
5461	O
,	O
6533	O
,	O
6541	O
,	O
6601	O
,	O
7777	O
,	O
8149	O
,	O
8401	O
,	O
11133	O
...	O
,	O
305	O
,	O
481	O
,	O
645	O
,	O
793	O
,	O
1729	O
,	O
2047	O
,	O
2257	O
,	O
2465	O
,	O
4577	O
,	O
4921	O
,	O
5041	O
,	O
5185	O
,	O
8113	O
,	O
1265	O
...	O
,	O
91	O
,	O
133	O
,	O
145	O
,	O
247	O
,	O
377	O
,	O
385	O
,	O
1649	O
,	O
1729	O
,	O
2041	O
,	O
2233	O
,	O
2465	O
,	O
2821	O
,	O
3553	O
,	O
6305	O
,	O
8911	O
,	O
9073	O
,	O
1321	O
...	O
,	O
85	O
,	O
105	O
,	O
561	O
,	O
1099	O
,	O
1785	O
,	O
2465	O
,	O
5149	O
,	O
5185	O
,	O
7107	O
,	O
8841	O
,	O
8911	O
,	O
9577	O
,	O
9637	O
,	O
1415	O
...	O
,	O
65	O
,	O
481	O
,	O
781	O
,	O
793	O
,	O
841	O
,	O
985	O
,	O
1541	O
,	O
2257	O
,	O
2465	O
,	O
2561	O
,	O
2743	O
,	O
3277	O
,	O
5185	O
,	O
5713	O
,	O
6533	O
,	O
6541	O
,	O
7171	O
,	O
7449	O
,	O
7585	O
,	O
8321	O
,	O
9073	O
,	O
15341	O
...	O
,	O
1477	O
,	O
1541	O
,	O
1687	O
,	O
1729	O
,	O
1921	O
,	O
3277	O
,	O
6541	O
,	O
9073	O
,	O
1615	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
179	O
...	O
,	O
91	O
,	O
145	O
,	O
781	O
,	O
1111	O
,	O
1305	O
,	O
1729	O
,	O
2149	O
,	O
2821	O
,	O
4033	O
,	O
4187	O
,	O
5365	O
,	O
5833	O
,	O
6697	O
,	O
7171	O
,	O
1825	O
...	O
,	O
49	O
,	O
65	O
,	O
133	O
,	O
325	O
,	O
343	O
,	O
425	O
,	O
1105	O
,	O
1225	O
,	O
1369	O
,	O
1387	O
,	O
1729	O
,	O
1921	O
,	O
2149	O
,	O
2465	O
,	O
2977	O
,	O
4577	O
,	O
5725	O
,	O
5833	O
,	O
5941	O
,	O
6305	O
,	O
6517	O
,	O
6601	O
,	O
7345	O
,	O
199	O
...	O
,	O
45	O
,	O
49	O
,	O
169	O
,	O
343	O
,	O
561	O
,	O
889	O
,	O
905	O
,	O
1105	O
,	O
1661	O
,	O
1849	O
,	O
2353	O
,	O
2465	O
,	O
2701	O
,	O
3201	O
,	O
4033	O
,	O
4681	O
,	O
5461	O
,	O
5713	O
,	O
6541	O
,	O
6697	O
,	O
7957	O
,	O
8145	O
,	O
8281	O
,	O
8401	O
,	O
9997	O
,	O
2021	O
...	O
,	O
57	O
,	O
133	O
,	O
671	O
,	O
889	O
,	O
1281	O
,	O
1653	O
,	O
1729	O
,	O
1891	O
,	O
2059	O
,	O
2413	O
,	O
2761	O
,	O
3201	O
,	O
5461	O
,	O
5473	O
,	O
5713	O
,	O
5833	O
,	O
6601	O
,	O
6817	O
,	O
7999	O
,	O
2165	O
...	O
,	O
221	O
,	O
703	O
,	O
793	O
,	O
1045	O
,	O
1105	O
,	O
2465	O
,	O
3781	O
,	O
5185	O
,	O
5473	O
,	O
6541	O
,	O
7363	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
485	O
,	O
1183	O
,	O
1247	O
,	O
1541	O
,	O
1729	O
,	O
2041	O
,	O
2047	O
,	O
2413	O
,	O
2465	O
,	O
2821	O
,	O
3241	O
,	O
3801	O
,	O
5551	O
,	O
7665	O
,	O
9453	O
,	O
2333	O
...	O
,	O
169	O
,	O
265	O
,	O
341	O
,	O
385	O
,	O
481	O
,	O
553	O
,	O
1065	O
,	O
1271	O
,	O
1729	O
,	O
2321	O
,	O
2465	O
,	O
2701	O
,	O
2821	O
,	O
3097	O
,	O
4033	O
,	O
4081	O
,	O
4345	O
,	O
4371	O
,	O
4681	O
,	O
5149	O
,	O
6533	O
,	O
6541	O
,	O
7189	O
,	O
7957	O
,	O
8321	O
,	O
8651	O
,	O
8745	O
,	O
8911	O
,	O
9805	O
,	O
2425	O
...	O
,	O
175	O
,	O
553	O
,	O
805	O
,	O
949	O
,	O
1541	O
,	O
1729	O
,	O
1825	O
,	O
1975	O
,	O
2413	O
,	O
2465	O
,	O
2701	O
,	O
3781	O
,	O
4537	O
,	O
6931	O
,	O
7501	O
,	O
9085	O
,	O
9361	O
,	O
25217	O
...	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
5731	O
,	O
6601	O
,	O
7449	O
,	O
7813	O
,	O
8029	O
,	O
8911	O
,	O
9881	O
,	O
269	O
...	O
,	O
25	O
,	O
27	O
,	O
45	O
,	O
133	O
,	O
217	O
,	O
225	O
,	O
475	O
,	O
561	O
,	O
589	O
,	O
703	O
,	O
925	O
,	O
1065	O
,	O
2465	O
,	O
3325	O
,	O
3385	O
,	O
3565	O
,	O
3825	O
,	O
4741	O
,	O
4921	O
,	O
5041	O
,	O
5425	O
,	O
6697	O
,	O
8029	O
,	O
9073	O
,	O
2765	O
...	O
,	O
121	O
,	O
133	O
,	O
259	O
,	O
341	O
,	O
365	O
,	O
481	O
,	O
703	O
,	O
1001	O
,	O
1541	O
,	O
1649	O
,	O
1729	O
,	O
1891	O
,	O
2465	O
,	O
2821	O
,	O
2981	O
,	O
2993	O
,	O
3281	O
,	O
4033	O
,	O
4745	O
,	O
4921	O
,	O
4961	O
,	O
5461	O
,	O
6305	O
,	O
6533	O
,	O
7381	O
,	O
7585	O
,	O
8321	O
,	O
8401	O
,	O
8911	O
,	O
9809	O
,	O
9841	O
,	O
9881	O
,	O
289	O
...	O
,	O
27	O
,	O
145	O
,	O
261	O
,	O
361	O
,	O
529	O
,	O
785	O
,	O
1305	O
,	O
1431	O
,	O
2041	O
,	O
2413	O
,	O
2465	O
,	O
3201	O
,	O
3277	O
,	O
4553	O
,	O
4699	O
,	O
5149	O
,	O
7065	O
,	O
8321	O
,	O
8401	O
,	O
9841	O
,	O
2915	O
...	O
,	O
21	O
,	O
91	O
,	O
105	O
,	O
341	O
,	O
469	O
,	O
481	O
,	O
793	O
,	O
871	O
,	O
1729	O
,	O
1897	O
,	O
2105	O
,	O
2257	O
,	O
2821	O
,	O
4371	O
,	O
4411	O
,	O
5149	O
,	O
5185	O
,	O
5473	O
,	O
5565	O
,	O
6097	O
,	O
7161	O
,	O
8321	O
,	O
8401	O
,	O
8421	O
,	O
8841	O
,	O
3049	O
...	O
,	O
133	O
,	O
217	O
,	O
341	O
,	O
403	O
,	O
469	O
,	O
589	O
,	O
637	O
,	O
871	O
,	O
901	O
,	O
931	O
,	O
1273	O
,	O
1537	O
,	O
1729	O
,	O
2059	O
,	O
2077	O
,	O
2821	O
,	O
3097	O
,	O
3277	O
,	O
4081	O
,	O
4097	O
,	O
5729	O
,	O
6031	O
,	O
6061	O
,	O
6097	O
,	O
6409	O
,	O
6817	O
,	O
7657	O
,	O
8023	O
,	O
8029	O
,	O
8401	O
,	O
9881	O
,	O
...	O
</s>
