Fermat pseudoprime

In number theory, the Fermat pseudoprimes make up the most important class of pseudoprimes that come from Fermat's little theorem.

Definition[edit]

Fermat's little theorem states that if p is prime and a is coprime to p, then a^p−1 − 1 is divisible by p. For an integer a > 1, if a composite integer x divides a^x−1 − 1, then x is called a Fermat pseudoprime to base a. In other words, a composite integer is a Fermat pseudoprime to base a if it successfully passes the Fermat primality test for the base a.^[1] It follows that if x is a Fermat pseudoprime to base a, then x is coprime to a. ^[2]

The smallest base-2 Fermat pseudoprime is 341. It is not a prime, since it equals 11·31, but it satisfies Fermat's little theorem: 2³⁴⁰ ≡ 1 (mod 341) and thus passes the
Fermat primality test for the base 2.

Pseudoprimes to base 2 are sometimes called Poulet numbers, after the Belgian mathematician Paul Poulet, Sarrus numbers, or Fermatians (sequence A001567 in the OEIS).

A Fermat pseudoprime is often called a pseudoprime, with the modifier Fermat being understood.

An integer x that is a Fermat pseudoprime for all values of a that are coprime to x is called a Carmichael number.^[1]

Variations[edit]

Some sources use variations of the definition, for example to only allow odd numbers to be pseudoprimes.^[3]

Every odd number q satisfies $a^{q-1}equiv 1{pmod {q}}$ for $a=q-1$ . This trivial case is excluded in the definition of a Fermat pseudoprime given by Crandall and Pomerance:^[4]

A composite number q is a Fermat pseudoprime to a base a, if

a q -1 \equiv 1 mod q

and

a \neq \pm1 mod q

.

Properties[edit]

Distribution[edit]

There are infinitely many pseudoprimes to a given base (in fact, infinitely many strong pseudoprimes (see Theorem 1 of
^[5])
and infinitely many Carmichael numbers
^[6])
, but they are rather rare.
There are only three pseudoprimes to base 2 below 1000, 245 below one million, and only 21853 less than 25·10⁹ (see Table 1 of ^[5]).

Starting at 17·257, the product of consecutive Fermat numbers is a base-2 pseudoprime, and so are all Fermat composite and Mersenne composite.

Factorizations[edit]

The factorizations of the 60 Poulet numbers up to 60787, including 13 Carmichael numbers (in bold), are in the following table.

(sequence A001567 in the OEIS)

Poulet 1 to 15
341	11 · 31
561	3 · 11 · 17
645	3 · 5 · 43
1105	5 · 13 · 17
1387	19 · 73
1729	7 · 13 · 19
1905	3 · 5 · 127
2047	23 · 89
2465	5 · 17 · 29
2701	37 · 73
2821	7 · 13 · 31
3277	29 · 113
4033	37 · 109
4369	17 · 257
4371	3 · 31 · 47

Poulet 16 to 30
4681	31 · 151
5461	43 · 127
6601	7 · 23 · 41
7957	73 · 109
8321	53 · 157
8481	3 · 11 · 257
8911	7 · 19 · 67
10261	31 · 331
10585	5 · 29 · 73
11305	5 · 7 · 17 · 19
12801	3 · 17 · 251
13741	7 · 13 · 151
13747	59 · 233
13981	11 · 31 · 41
14491	43 · 337

Poulet 31 to 45
15709	23 · 683
15841	7 · 31 · 73
16705	5 · 13 · 257
18705	3 · 5 · 29 · 43
18721	97 · 193
19951	71 · 281
23001	3 · 11 · 17 · 41
23377	97 · 241
25761	3 · 31 · 277
29341	13 · 37 · 61
30121	7 · 13 · 331
30889	17 · 23 · 79
31417	89 · 353
31609	73 · 433
31621	103 · 307

Poulet 46 to 60
33153	3 · 43 · 257
34945	5 · 29 · 241
35333	89 · 397
39865	5 · 7 · 17 · 67
41041	7 · 11 · 13 · 41
41665	5 · 13 · 641
42799	127 · 337
46657	13 · 37 · 97
49141	157 · 313
49981	151 · 331
52633	7 · 73 · 103
55245	3 · 5 · 29 · 127
57421	7 · 13 · 631
60701	101 · 601
60787	89 · 683

A Poulet number all of whose divisors d divide 2^d − 2 is called a super-Poulet number. There are infinitely many Poulet numbers which are not super-Poulet Numbers.^[7]

Smallest Fermat pseudoprimes[edit]

The smallest pseudoprime for each base a ≤ 200 is given in the following table; the colors mark the number of prime factors. Unlike in the definition at the start of the article, pseudoprimes below a are excluded in the table. (For that to allow pseudoprimes below a, see OEIS: A090086)

(sequence A007535 in the OEIS)

a	smallest p-p	a	smallest p-p	a	smallest p-p	a	smallest p-p
1	4 = 2²	51	65 = 5 · 13	101	175 = 5² · 7	151	175 = 5² · 7
2	341 = 11 · 31	52	85 = 5 · 17	102	133 = 7 · 19	152	153 = 3² · 17
3	91 = 7 · 13	53	65 = 5 · 13	103	133 = 7 · 19	153	209 = 11 · 19
4	15 = 3 · 5	54	55 = 5 · 11	104	105 = 3 · 5 · 7	154	155 = 5 · 31
5	124 = 2² · 31	55	63 = 3² · 7	105	451 = 11 · 41	155	231 = 3 · 7 · 11
6	35 = 5 · 7	56	57 = 3 · 19	106	133 = 7 · 19	156	217 = 7 · 31
7	25 = 5²	57	65 = 5 · 13	107	133 = 7 · 19	157	186 = 2 · 3 · 31
8	9 = 3²	58	133 = 7 · 19	108	341 = 11 · 31	158	159 = 3 · 53
9	28 = 2² · 7	59	87 = 3 · 29	109	117 = 3² · 13	159	247 = 13 · 19
10	33 = 3 · 11	60	341 = 11 · 31	110	111 = 3 · 37	160	161 = 7 · 23
11	15 = 3 · 5	61	91 = 7 · 13	111	190 = 2 · 5 · 19	161	190 = 2 · 5 · 19
12	65 = 5 · 13	62	63 = 3² · 7	112	121 = 11²	162	481 = 13 · 37
13	21 = 3 · 7	63	341 = 11 · 31	113	133 = 7 · 19	163	186 = 2 · 3 · 31
14	15 = 3 · 5	64	65 = 5 · 13	114	115 = 5 · 23	164	165 = 3 · 5 · 11
15	341 = 11 · 31	65	112 = 2⁴ · 7	115	133 = 7 · 19	165	172 = 2² · 43
16	51 = 3 · 17	66	91 = 7 · 13	116	117 = 3² · 13	166	301 = 7 · 43
17	45 = 3² · 5	67	85 = 5 · 17	117	145 = 5 · 29	167	231 = 3 · 7 · 11
18	25 = 5²	68	69 = 3 · 23	118	119 = 7 · 17	168	169 = 13²
19	45 = 3² · 5	69	85 = 5 · 17	119	177 = 3 · 59	169	231 = 3 · 7 · 11
20	21 = 3 · 7	70	169 = 13²	120	121 = 11²	170	171 = 3² · 19
21	55 = 5 · 11	71	105 = 3 · 5 · 7	121	133 = 7 · 19	171	215 = 5 · 43
22	69 = 3 · 23	72	85 = 5 · 17	122	123 = 3 · 41	172	247 = 13 · 19
23	33 = 3 · 11	73	111 = 3 · 37	123	217 = 7 · 31	173	205 = 5 · 41
24	25 = 5²	74	75 = 3 · 5²	124	125 = 5³	174	175 = 5² · 7
25	28 = 2² · 7	75	91 = 7 · 13	125	133 = 7 · 19	175	319 = 11 · 19
26	27 = 3³	76	77 = 7 · 11	126	247 = 13 · 19	176	177 = 3 · 59
27	65 = 5 · 13	77	247 = 13 · 19	127	153 = 3² · 17	177	196 = 2² · 7²
28	45 = 3² · 5	78	341 = 11 · 31	128	129 = 3 · 43	178	247 = 13 · 19
29	35 = 5 · 7	79	91 = 7 · 13	129	217 = 7 · 31	179	185 = 5 · 37
30	49 = 7²	80	81 = 3⁴	130	217 = 7 · 31	180	217 = 7 · 31
31	49 = 7²	81	85 = 5 · 17	131	143 = 11 · 13	181	195 = 3 · 5 · 13
32	33 = 3 · 11	82	91 = 7 · 13	132	133 = 7 · 19	182	183 = 3 · 61
33	85 = 5 · 17	83	105 = 3 · 5 · 7	133	145 = 5 · 29	183	221 = 13 · 17
34	35 = 5 · 7	84	85 = 5 · 17	134	135 = 3³ · 5	184	185 = 5 · 37
35	51 = 3 · 17	85	129 = 3 · 43	135	221 = 13 · 17	185	217 = 7 · 31
36	91 = 7 · 13	86	87 = 3 · 29	136	265 = 5 · 53	186	187 = 11 · 17
37	45 = 3² · 5	87	91 = 7 · 13	137	148 = 2² · 37	187	217 = 7 · 31
38	39 = 3 · 13	88	91 = 7 · 13	138	259 = 7 · 37	188	189 = 3³ · 7
39	95 = 5 · 19	89	99 = 3² · 11	139	161 = 7 · 23	189	235 = 5 · 47
40	91 = 7 · 13	90	91 = 7 · 13	140	141 = 3 · 47	190	231 = 3 · 7 · 11
41	105 = 3 · 5 · 7	91	115 = 5 · 23	141	355 = 5 · 71	191	217 = 7 · 31
42	205 = 5 · 41	92	93 = 3 · 31	142	143 = 11 · 13	192	217 = 7 · 31
43	77 = 7 · 11	93	301 = 7 · 43	143	213 = 3 · 71	193	276 = 2² · 3 · 23
44	45 = 3² · 5	94	95 = 5 · 19	144	145 = 5 · 29	194	195 = 3 · 5 · 13
45	76 = 2² · 19	95	141 = 3 · 47	145	153 = 3² · 17	195	259 = 7 · 37
46	133 = 7 · 19	96	133 = 7 · 19	146	147 = 3 · 7²	196	205 = 5 · 41
47	65 = 5 · 13	97	105 = 3 · 5 · 7	147	169 = 13²	197	231 = 3 · 7 · 11
48	49 = 7²	98	99 = 3² · 11	148	231 = 3 · 7 · 11	198	247 = 13 · 19
49	66 = 2 · 3 · 11	99	145 = 5 · 29	149	175 = 5² · 7	199	225 = 3² · 5²
50	51 = 3 · 17	100	153 = 3² · 17	150	169 = 13²	200	201 = 3 · 67

List of Fermat pseudoprimes in fixed base n[edit]

n	First few Fermat pseudoprimes in base n	OEIS sequence
1	4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, ... (All composites)	A002808
2	341, 561, 645, 1105, 1387, 1729, 1905, 2047, 2465, 2701, 2821, 3277, 4033, 4369, 4371, 4681, 5461, 6601, 7957, 8321, 8481, 8911, ...	A001567
3	91, 121, 286, 671, 703, 949, 1105, 1541, 1729, 1891, 2465, 2665, 2701, 2821, 3281, 3367, 3751, 4961, 5551, 6601, 7381, 8401, 8911, ...	A005935
4	15, 85, 91, 341, 435, 451, 561, 645, 703, 1105, 1247, 1271, 1387, 1581, 1695, 1729, 1891, 1905, 2047, 2071, 2465, 2701, 2821, 3133, 3277, 3367, 3683, 4033, 4369, 4371, 4681, 4795, 4859, 5461, 5551, 6601, 6643, 7957, 8321, 8481, 8695, 8911, 9061, 9131, 9211, 9605, 9919, ...	A020136
5	4, 124, 217, 561, 781, 1541, 1729, 1891, 2821, 4123, 5461, 5611, 5662, 5731, 6601, 7449, 7813, 8029, 8911, 9881, ...	A005936
6	35, 185, 217, 301, 481, 1105, 1111, 1261, 1333, 1729, 2465, 2701, 2821, 3421, 3565, 3589, 3913, 4123, 4495, 5713, 6533, 6601, 8029, 8365, 8911, 9331, 9881, ...	A005937
7	6, 25, 325, 561, 703, 817, 1105, 1825, 2101, 2353, 2465, 3277, 4525, 4825, 6697, 8321, ...	A005938
8	9, 21, 45, 63, 65, 105, 117, 133, 153, 231, 273, 341, 481, 511, 561, 585, 645, 651, 861, 949, 1001, 1105, 1281, 1365, 1387, 1417, 1541, 1649, 1661, 1729, 1785, 1905, 2047, 2169, 2465, 2501, 2701, 2821, 3145, 3171, 3201, 3277, 3605, 3641, 4005, 4033, 4097, 4369, 4371, 4641, 4681, 4921, 5461, 5565, 5963, 6305, 6533, 6601, 6951, 7107, 7161, 7957, 8321, 8481, 8911, 9265, 9709, 9773, 9881, 9945, ...	A020137
9	4, 8, 28, 52, 91, 121, 205, 286, 364, 511, 532, 616, 671, 697, 703, 946, 949, 1036, 1105, 1288, 1387, 1541, 1729, 1891, 2465, 2501, 2665, 2701, 2806, 2821, 2926, 3052, 3281, 3367, 3751, 4376, 4636, 4961, 5356, 5551, 6364, 6601, 6643, 7081, 7381, 7913, 8401, 8695, 8744, 8866, 8911, ...	A020138
10	9, 33, 91, 99, 259, 451, 481, 561, 657, 703, 909, 1233, 1729, 2409, 2821, 2981, 3333, 3367, 4141, 4187, 4521, 5461, 6533, 6541, 6601, 7107, 7471, 7777, 8149, 8401, 8911, ...	A005939
11	10, 15, 70, 133, 190, 259, 305, 481, 645, 703, 793, 1105, 1330, 1729, 2047, 2257, 2465, 2821, 4577, 4921, 5041, 5185, 6601, 7869, 8113, 8170, 8695, 8911, 9730, ...	A020139
12	65, 91, 133, 143, 145, 247, 377, 385, 703, 1045, 1099, 1105, 1649, 1729, 1885, 1891, 2041, 2233, 2465, 2701, 2821, 2983, 3367, 3553, 5005, 5365, 5551, 5785, 6061, 6305, 6601, 8911, 9073, ...	A020140
13	4, 6, 12, 21, 85, 105, 231, 244, 276, 357, 427, 561, 1099, 1785, 1891, 2465, 2806, 3605, 5028, 5149, 5185, 5565, 6601, 7107, 8841, 8911, 9577, 9637, ...	A020141
14	15, 39, 65, 195, 481, 561, 781, 793, 841, 985, 1105, 1111, 1541, 1891, 2257, 2465, 2561, 2665, 2743, 3277, 5185, 5713, 6501, 6533, 6541, 7107, 7171, 7449, 7543, 7585, 8321, 9073, ...	A020142
15	14, 341, 742, 946, 1477, 1541, 1687, 1729, 1891, 1921, 2821, 3133, 3277, 4187, 6541, 6601, 7471, 8701, 8911, 9073, ...	A020143
16	15, 51, 85, 91, 255, 341, 435, 451, 561, 595, 645, 703, 1105, 1247, 1261, 1271, 1285, 1387, 1581, 1687, 1695, 1729, 1891, 1905, 2047, 2071, 2091, 2431, 2465, 2701, 2821, 3133, 3277, 3367, 3655, 3683, 4033, 4369, 4371, 4681, 4795, 4859, 5083, 5151, 5461, 5551, 6601, 6643, 7471, 7735, 7957, 8119, 8227, 8245, 8321, 8481, 8695, 8749, 8911, 9061, 9131, 9211, 9605, 9919, ...	A020144
17	4, 8, 9, 16, 45, 91, 145, 261, 781, 1111, 1228, 1305, 1729, 1885, 2149, 2821, 3991, 4005, 4033, 4187, 4912, 5365, 5662, 5833, 6601, 6697, 7171, 8481, 8911, ...	A020145
18	25, 49, 65, 85, 133, 221, 323, 325, 343, 425, 451, 637, 931, 1105, 1225, 1369, 1387, 1649, 1729, 1921, 2149, 2465, 2701, 2821, 2825, 2977, 3325, 4165, 4577, 4753, 5525, 5725, 5833, 5941, 6305, 6517, 6601, 7345, 8911, 9061, ...	A020146
19	6, 9, 15, 18, 45, 49, 153, 169, 343, 561, 637, 889, 905, 906, 1035, 1105, 1629, 1661, 1849, 1891, 2353, 2465, 2701, 2821, 2955, 3201, 4033, 4681, 5461, 5466, 5713, 6223, 6541, 6601, 6697, 7957, 8145, 8281, 8401, 8869, 9211, 9997, ...	A020147
20	21, 57, 133, 231, 399, 561, 671, 861, 889, 1281, 1653, 1729, 1891, 2059, 2413, 2501, 2761, 2821, 2947, 3059, 3201, 4047, 5271, 5461, 5473, 5713, 5833, 6601, 6817, 7999, 8421, 8911, ...	A020148
21	4, 10, 20, 55, 65, 85, 221, 703, 793, 1045, 1105, 1852, 2035, 2465, 3781, 4630, 5185, 5473, 5995, 6541, 7363, 8695, 8965, 9061, ...	A020149
22	21, 69, 91, 105, 161, 169, 345, 483, 485, 645, 805, 1105, 1183, 1247, 1261, 1541, 1649, 1729, 1891, 2037, 2041, 2047, 2413, 2465, 2737, 2821, 3241, 3605, 3801, 5551, 5565, 5963, 6019, 6601, 6693, 7081, 7107, 7267, 7665, 8119, 8365, 8421, 8911, 9453, ...	A020150
23	22, 33, 91, 154, 165, 169, 265, 341, 385, 451, 481, 553, 561, 638, 946, 1027, 1045, 1065, 1105, 1183, 1271, 1729, 1738, 1749, 2059, 2321, 2465, 2501, 2701, 2821, 2926, 3097, 3445, 4033, 4081, 4345, 4371, 4681, 5005, 5149, 6253, 6369, 6533, 6541, 7189, 7267, 7957, 8321, 8365, 8651, 8745, 8911, 8965, 9805, ...	A020151
24	25, 115, 175, 325, 553, 575, 805, 949, 1105, 1541, 1729, 1771, 1825, 1975, 2413, 2425, 2465, 2701, 2737, 2821, 2885, 3781, 4207, 4537, 6601, 6931, 6943, 7081, 7189, 7471, 7501, 7813, 8725, 8911, 9085, 9361, 9809, ...	A020152
25	4, 6, 8, 12, 24, 28, 39, 66, 91, 124, 217, 232, 276, 403, 426, 451, 532, 561, 616, 703, 781, 804, 868, 946, 1128, 1288, 1541, 1729, 1891, 2047, 2701, 2806, 2821, 2911, 2926, 3052, 3126, 3367, 3592, 3976, 4069, 4123, 4207, 4564, 4636, 4686, 5321, 5461, 5551, 5611, 5662, 5731, 5963, 6601, 7449, 7588, 7813, 8029, 8646, 8911, 9881, 9976, ...	A020153
26	9, 15, 25, 27, 45, 75, 133, 135, 153, 175, 217, 225, 259, 425, 475, 561, 589, 675, 703, 775, 925, 1035, 1065, 1147, 2465, 3145, 3325, 3385, 3565, 3825, 4123, 4525, 4741, 4921, 5041, 5425, 6093, 6475, 6525, 6601, 6697, 8029, 8695, 8911, 9073, ...	A020154
27	26, 65, 91, 121, 133, 247, 259, 286, 341, 365, 481, 671, 703, 949, 1001, 1105, 1541, 1649, 1729, 1891, 2071, 2465, 2665, 2701, 2821, 2981, 2993, 3146, 3281, 3367, 3605, 3751, 4033, 4745, 4921, 4961, 5299, 5461, 5551, 5611, 5621, 6305, 6533, 6601, 7381, 7585, 7957, 8227, 8321, 8401, 8911, 9139, 9709, 9809, 9841, 9881, 9919, ...	A020155
28	9, 27, 45, 87, 145, 261, 361, 529, 561, 703, 783, 785, 1105, 1305, 1413, 1431, 1885, 2041, 2413, 2465, 2871, 3201, 3277, 4553, 4699, 5149, 5181, 5365, 7065, 8149, 8321, 8401, 9841, ...	A020156
29	4, 14, 15, 21, 28, 35, 52, 91, 105, 231, 268, 341, 364, 469, 481, 561, 651, 793, 871, 1105, 1729, 1876, 1897, 2105, 2257, 2821, 3484, 3523, 4069, 4371, 4411, 5149, 5185, 5356, 5473, 5565, 5611, 6097, 6601, 7161, 7294, 8321, 8401, 8421, 8841, 8911, ...	A020157
30	49, 91, 133, 217, 247, 341, 403, 469, 493, 589, 637, 703, 871, 899, 901, 931, 1273, 1519, 1537, 1729, 2059, 2077, 2821, 3097, 3277, 3283, 3367, 3577, 4081, 4097, 4123, 5729, 6031, 6061, 6097, 6409, 6601, 6817, 7657, 8023, 8029, 8401, 8911, 9881, ...	A020158

For more information (base 31 to 100), see OEIS: A020159 to OEIS: A020228, and for all bases up to 150, see table of Fermat pseudoprimes (text in German), this page does not define n is a pseudoprime to a base congruent to 1 or -1 (mod n)

Which bases b make n a Fermat pseudoprime?[edit]

If composite $n$ is even, then $n$ is a Fermat pseudoprime to the trivial bases ${displaystyle bequiv 1{pmod {n}}}$ .
If composite $n$ is odd, then $n$ is a Fermat pseudoprime to the trivial bases ${displaystyle bequiv pm 1{pmod {n}}}$ .

For any composite $n$ , the number of distinct bases $b$ modulo $n$ , for which $n$ is a Fermat pseudoprime base $b$ , is
^[8]^{:Thm. 1, p. 1392}

{displaystyle prod _{i=1}^{k}gcd(p_{i}-1,n-1)}

where ${displaystyle p_{1},dots ,p_{k}}$ are the distinct prime factors of $n$ . This includes the trivial bases.

For example, for ${displaystyle n=341=11cdot 31}$ , this product is ${displaystyle gcd(10,340)cdot gcd(30,340)=100}$ . For ${displaystyle n=341}$ , the smallest such nontrivial base is $b=2$ .

Every odd composite $n$ is a Fermat pseudoprime to at least two nontrivial bases modulo $n$ unless $n$ is a power of 3.^[8]^{:Cor. 1, p. 1393}

For composite n < 200, the following is a table of all bases b < n which n is a Fermat pseudoprime. If a composite number n is not in the table (or n is in the sequence A209211), then n is a pseudoprime only to the trivial base 1 modulo n.

n	bases b to which n is a Fermat pseudoprime (< n)	number of the bases of b (< n) (sequence A063994 in the OEIS)
9	1, 8	2
15	1, 4, 11, 14	4
21	1, 8, 13, 20	4
25	1, 7, 18, 24	4
27	1, 26	2
28	1, 9, 25	3
33	1, 10, 23, 32	4
35	1, 6, 29, 34	4
39	1, 14, 25, 38	4
45	1, 8, 17, 19, 26, 28, 37, 44	8
49	1, 18, 19, 30, 31, 48	6
51	1, 16, 35, 50	4
52	1, 9, 29	3
55	1, 21, 34, 54	4
57	1, 20, 37, 56	4
63	1, 8, 55, 62	4
65	1, 8, 12, 14, 18, 21, 27, 31, 34, 38, 44, 47, 51, 53, 57, 64	16
66	1, 25, 31, 37, 49	5
69	1, 22, 47, 68	4
70	1, 11, 51	3
75	1, 26, 49, 74	4
76	1, 45, 49	3
77	1, 34, 43, 76	4
81	1, 80	2
85	1, 4, 13, 16, 18, 21, 33, 38, 47, 52, 64, 67, 69, 72, 81, 84	16
87	1, 28, 59, 86	4
91	1, 3, 4, 9, 10, 12, 16, 17, 22, 23, 25, 27, 29, 30, 36, 38, 40, 43, 48, 51, 53, 55, 61, 62, 64, 66, 68, 69, 74, 75, 79, 81, 82, 87, 88, 90	36
93	1, 32, 61, 92	4
95	1, 39, 56, 94	4
99	1, 10, 89, 98	4
105	1, 8, 13, 22, 29, 34, 41, 43, 62, 64, 71, 76, 83, 92, 97, 104	16
111	1, 38, 73, 110	4
112	1, 65, 81	3
115	1, 24, 91, 114	4
117	1, 8, 44, 53, 64, 73, 109, 116	8
119	1, 50, 69, 118	4
121	1, 3, 9, 27, 40, 81, 94, 112, 118, 120	10
123	1, 40, 83, 122	4
124	1, 5, 25	3
125	1, 57, 68, 124	4
129	1, 44, 85, 128	4
130	1, 61, 81	3
133	1, 8, 11, 12, 18, 20, 26, 27, 30, 31, 37, 39, 45, 46, 50, 58, 64, 65, 68, 69, 75, 83, 87, 88, 94, 96, 102, 103, 106, 107, 113, 115, 121, 122, 125, 132	36
135	1, 26, 109, 134	4
141	1, 46, 95, 140	4
143	1, 12, 131, 142	4
145	1, 12, 17, 28, 41, 46, 57, 59, 86, 88, 99, 104, 117, 128, 133, 144	16
147	1, 50, 97, 146	4
148	1, 121, 137	3
153	1, 8, 19, 26, 35, 53, 55, 64, 89, 98, 100, 118, 127, 134, 145, 152	16
154	1, 23, 67	3
155	1, 61, 94, 154	4
159	1, 52, 107, 158	4
161	1, 22, 139, 160	4
165	1, 23, 32, 34, 43, 56, 67, 76, 89, 98, 109, 122, 131, 133, 142, 164	16
169	1, 19, 22, 23, 70, 80, 89, 99, 146, 147, 150, 168	12
171	1, 37, 134, 170	4
172	1, 49, 165	3
175	1, 24, 26, 51, 74, 76, 99, 101, 124, 149, 151, 174	12
176	1, 49, 81, 97, 113	5
177	1, 58, 119, 176	4
183	1, 62, 121, 182	4
185	1, 6, 31, 36, 38, 43, 68, 73, 112, 117, 142, 147, 149, 154, 179, 184	16
186	1, 97, 109, 157, 163	5
187	1, 67, 120, 186	4
189	1, 55, 134, 188	4
190	1, 11, 61, 81, 101, 111, 121, 131, 161	9
195	1, 14, 64, 79, 116, 131, 181, 194	8
196	1, 165, 177	3

For more information (n = 201 to 5000), see,^[9] this page does not define n is a pseudoprime to a base congruent to 1 or -1 (mod n). Note that when p is a prime, p² is a Fermat pseudoprime to base b if and only if p is a Wieferich prime to base b. For example, 1093² = 1194649 is a Fermat pseudoprime to base 2, and 11² = 121 is a Fermat pseudoprime to base 3.

The number of the values of b for n are (For n prime, the number of the values of b must be n - 1, since all b satisfy the Fermat little theorem)

1, 1, 2, 1, 4, 1, 6, 1, 2, 1, 10, 1, 12, 1, 4, 1, 16, 1, 18, 1, 4, 1, 22, 1, 4, 1, 2, 3, 28, 1, 30, 1, 4, 1, 4, 1, 36, 1, 4, 1, 40, 1, 42, 1, 8, 1, 46, 1, 6, 1, ... (sequence A063994 in the OEIS)

The least base b > 1 which n is a pseudoprime to base b (or prime number) are

2, 3, 2, 5, 2, 7, 2, 9, 8, 11, 2, 13, 2, 15, 4, 17, 2, 19, 2, 21, 8, 23, 2, 25, 7, 27, 26, 9, 2, 31, 2, 33, 10, 35, 6, 37, 2, 39, 14, 41, 2, 43, 2, 45, 8, 47, 2, 49, 18, 51, ... (sequence A105222 in the OEIS)

The number of the values of b for n must divides $varphi$ (n), or A000010(n) = 1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, 12, 6, 8, 8, 16, 6, 18, 8, 12, 10, 22, 8, 20, 12, 18, 12, 28, 8, 30, 16, 20, 16, 24, 12, 36, 18, 24, 16, 40, 12, 42, 20, 24, 22, 46, 16, 42, 20, ... (The quotient can be any natural number, and the quotient = 1 if and only if n is a prime or a Carmichael number (561, 1105, 1729, 2465, 2821, 6601, 8911, 10585, 15841, ... A002997), the quotient = 2 if and only if n is in the sequence: 4, 6, 15, 91, 703, 1891, 2701, 11305, 12403, 13981, 18721, ... A191311)

The least number with n values of b are (or 0 if no such number exists)

1, 3, 28, 5, 66, 7, 232, 45, 190, 11, 276, 13, 1106, 0, 286, 17, 1854, 19, 3820, 891, 2752, 23, 1128, 595, 2046, 0, 532, 29, 1770, 31, 9952, 425, 1288, 0, 2486, 37, 8474, 0, 742, 41, 3486, 43, 7612, 5589, 2356, 47, 13584, 325, 9850, 0, ... (sequence A064234 in the OEIS) (if and only if n is even and not totient of squarefree number, then the nth term of this sequence is 0)

Weak pseudoprimes[edit]

A composite number n which satisfy that ${displaystyle b^{n}equiv b{pmod {n}}}$ is called weak pseudoprime to base b. A pseudoprime to base a (under the usual definition) satisfies this condition. Conversely, a weak pseudoprime that's coprime with the base is a pseudoprime in the usual sense, otherwise this may or may not be the case. ^[10] The least weak pseudoprime to base b = 1, 2, ... are:

4, 341, 6, 4, 4, 6, 6, 4, 4, 6, 10, 4, 4, 14, 6, 4, 4, 6, 6, 4, 4, 6, 22, 4, 4, 9, 6, 4, 4, 6, 6, 4, 4, 6, 9, 4, 4, 38, 6, 4, 4, 6, 6, 4, 4, 6, 46, 4, 4, 10, ... (sequence A000790 in the OEIS)

Note that all terms are less than or equal to the smallest Carmichael number, 561. Except for 561, only semiprimes can occur in the above sequence, but not all semiprimes less than 561 occur, a semiprime pq (p ≤ q) less than 561 occurs in the above sequences if and only if p − 1 divides q − 1. (see OEIS: A108574) Besides, the smallest pseudoprime to base n (also not necessary exceeding n) (OEIS: A090086) is also usually semiprime, the first counterexample is A090086(648) = 385 = 5 × 7 × 11.

If we require n > b, they are (for b = 1, 2, ...)

4, 341, 6, 6, 10, 10, 14, 9, 12, 15, 15, 22, 21, 15, 21, 20, 34, 25, 38, 21, 28, 33, 33, 25, 28, 27, 39, 36, 35, 49, 49, 33, 44, 35, 45, 42, 45, 39, 57, 52, 82, 66, 77, 45, 55, 69, 65, 49, 56, 51, ... (sequence A239293 in the OEIS)

Carmichael numbers are weak pseudoprimes to all bases.

The smallest even weak pseudoprime in base 2 is 161038 (see OEIS: A006935).

Euler–Jacobi pseudoprimes[edit]

Another approach is to use more refined notions of pseudoprimality, e.g. strong pseudoprimes or Euler–Jacobi pseudoprimes, for which there are no analogues of Carmichael numbers. This leads to probabilistic algorithms such as the Solovay–Strassen primality test, the Baillie-PSW primality test, and the Miller–Rabin primality test, which produce what are known as industrial-grade primes. Industrial-grade primes are integers for which primality has not been "certified" (i.e. rigorously proven), but have undergone a test such as the Miller–Rabin test which has nonzero, but arbitrarily low, probability of failure.

Fermat primality test[edit]

Like the Lucas primality test, for a large number n (we assume that n is odd and nonsquare, or we can know that n is composite), we can choose the smallest positive integer b such that ${displaystyle left({tfrac {b}{n}}right)=-1}$ (where ${displaystyle left({tfrac {b}{n}}right)}$ is the Jacobi symbol) as the base and use Fermat test, Euler-Jacobi test or Miller-Rabin test to test the primality of n. (If b and n have a prime factor in common, then ${displaystyle left({tfrac {b}{n}}right)=0}$ , and if we found such b, then we can know that n is composite).

(This search will never succeed if n is square, and conversely if it does succeed, that is proof that n is not square. Thus, some time can be saved by delaying testing n for squareness until after the first few search steps have all failed.)

When use this test, the first few Fermat pseudoprimes are

217, 341, 561, 645, 703, 1105, 1387, 1729, 1825, 2465, 2701, 2821, 3277, 3281, 3367, 3751, 4371, 4961, 5461, 5551, 6601, 7957, 8911, 10225, 10261, 10585, 11521, 12025, 13741, 13747, 13981, 14089, 14383, 14491, 15709, 15841, 16297, 16471, 18721, 20425, 22945, 24727, 29341, 30673, 30857, 31621, 31697, 32791, 35333, 35425, 38503, 39865, 41041, 41329, 44287, 46657, 46999, 49105, 49141, 49321, 49981, ...

The first few Euler pseudoprimes are

217, 341, 561, 703, 1729, 1825, 2465, 3277, 3281, 4961, 5461, 8911, 10225, 10261, 11521, 12025, 14089, 15709, 15841, 16297, 20425, 29341, 30673, 30857, 31621, 31697, 35425, 39865, 41041, 41329, 44287, 46657, 49105, 49141, 49321, 49981, 50881, 54145, 65077, 68401, 72041, 75241, 75361, 80581, 83333, 88357, 88705, 96049, 96985, 97567, 97921, ...

The first few Euler-Jacobi pseudoprimes are

703, 3277, 3281, 8911, 14089, 29341, 44287, 49141, 80581, 88357, 97567, 102311, 104653, 121463, 152551, 172369, 173951, 182527, 188191, 195313, 196093, 216457, 218791, 304921, 313447, 314821, 320167, 364231, 410041, 458989, 476971, 489997, 491209, 497503, 658711, 665281, 721801, 777961, 800605, 838861, 859951, 873181, 877099, 973241, ...

The first few strong pseudoprimes are

703, 3277, 3281, 8911, 14089, 29341, 44287, 49141, 80581, 88357, 97567, 102311, 104653, 121463, 152551, 172369, 173951, 182527, 188191, 195313, 196093, 216457, 218791, 304921, 313447, 314821, 320167, 364231, 410041, 458989, 476971, 489997, 491209, 497503, 658711, 665281, 721801, 777961, 800605, 838861, 859951, 873181, 877099, 973241, ...

Test	Fermat primality test	Lucas primality test
Condition of the number n	n is odd and nonsquare
Choose base b (for Fermat) or Lucas parameters (P, Q) (for Lucas)	b is the first number in the sequence 2, 3, 4, 5, 6, 7, 8, ... such that ${displaystyle left({tfrac {b}{n}}right)=-1}$ (where ${displaystyle left({tfrac {b}{n}}right)}$ is the Jacobi symbol)	P = 1, Q is the first number in the sequence −1, 2, −2, 3, −3, 4, −4, ... such that ${displaystyle left({tfrac {1-4Q}{n}}right)=-1}$ (where ${displaystyle left({tfrac {1-4Q}{n}}right)}$ is the Jacobi symbol)
The first few pseudoprimes	217, 341, 561, 645, 703, 1105, 1387, 1729, 1825, 2465, 2701, 2821, 3277, 3281, 3367, 3751, 4371, 4961, 5461, 5551, 6601, 7957, 8911	323, 377, 1159, 1829, 3827, 5459, 5777, 9071, 9179
The first few strong pseudoprimes	703, 3277, 3281, 8911, 14089, 29341, 44287, 49141, 80581, 88357, 97567	5459, 5777, 10877, 16109, 18971, 22499, 24569, 25199, 40309, 58519, 75077, 97439

It is conjectured that there is no composite number that passes both the strong (Fermat) primality test and the strong Lucas primality test.

Applications[edit]

The rarity of such pseudoprimes has important practical implications. For example, public-key cryptography algorithms such as RSA require the ability to quickly find large primes. The usual algorithm to generate prime numbers is to generate random odd numbers and test them for primality. However, deterministic primality tests are slow. If the user is willing to tolerate an arbitrarily small chance that the number found is not a prime number but a pseudoprime, it is possible to use the much faster and simpler Fermat primality test.

References[edit]

^ ^a^b Desmedt, Yvo (2010). "Encryption Schemes". In Atallah, Mikhail J.; Blanton, Marina. Algorithms and theory of computation handbook: Special topics and techniques. CRC Press. pp. 10–23. ISBN 978-1-58488-820-8..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output q{quotes:"""""""'""'"}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}

^ "Contrapositive of this statement on math.stackexchange".

^ Weisstein, Eric W. "Fermat Pseudoprime". MathWorld.

^ Crandall, Richard; Pomerance, Carl (2001). "Theorem 3.4.2". Prime Numbers – A Computational Perspective. Springer-Verlag. p. 132.

^ ^a^b Pomerance, Carl; Selfridge, John L.; Wagstaff, Samuel S., Jr. (July 1980). "The pseudoprimes to 25·10⁹" (PDF). Mathematics of Computation. 35 (151): 1003–1026. doi:10.1090/S0025-5718-1980-0572872-7.

^ Alford, W. R.; Granville, Andrew; Pomerance, Carl (1994). "There are Infinitely Many Carmichael Numbers" (PDF). Annals of Mathematics. 140: 703–722. doi:10.2307/2118576.

^ Sierpinski, W. (1988-02-15), "Chapter V.7", in Ed. A. Schinzel, Elementary Theory of Numbers, North-Holland Mathematical Library (2 Sub ed.), Amsterdam: North Holland, p. 232, ISBN 9780444866622

^ ^a^b Robert Baillie; Samuel S. Wagstaff Jr. (October 1980). "Lucas Pseudoprimes" (PDF). Mathematics of Computation. 35 (152): 1391–1417. doi:10.1090/S0025-5718-1980-0583518-6. MR 0583518.

^ "Pseudoprimzahlen: Tabelle Pseudoprimzahlen (15 - 4999) – Wikibooks, Sammlung freier Lehr-, Sach- und Fachbücher". de.m.wikibooks.org. Retrieved 21 April 2018.

^ Michon, Gerard. "Pseudo-primes, Weak Pseudoprimes, Strong Pseudoprimes, Primality - Numericana". www.numericana.com. Retrieved 21 April 2018.

External links[edit]

W. F. Galway and Jan Feitsma, Tables of pseudoprimes and related data (comprehensive list of all pseudoprimes below 2⁶⁴, including factorization, strong pseudoprimes, and Carmichael numbers)

A research for pseudoprime

[desmedt-10-23-1] Desmedt, Yvo (2010). "Encryption Schemes". In Atallah, Mikhail J.; Blanton, Marina. Algorithms and theory of computation handbook: Special topics and techniques. CRC Press. pp. 10–23. ISBN 978-1-58488-820-8..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output q{quotes:"""""""'""'"}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}

[2] "Contrapositive of this statement on math.stackexchange".

[3] Weisstein, Eric W. "Fermat Pseudoprime". MathWorld.

[4] Crandall, Richard; Pomerance, Carl (2001). "Theorem 3.4.2". Prime Numbers – A Computational Perspective. Springer-Verlag. p. 132.

[PSW-5] Pomerance, Carl; Selfridge, John L.; Wagstaff, Samuel S., Jr. (July 1980). "The pseudoprimes to 25·10⁹" (PDF). Mathematics of Computation. 35 (151): 1003–1026. doi:10.1090/S0025-5718-1980-0572872-7.

[Alford1994-6] Alford, W. R.; Granville, Andrew; Pomerance, Carl (1994). "There are Infinitely Many Carmichael Numbers" (PDF). Annals of Mathematics. 140: 703–722. doi:10.2307/2118576.

[7] Sierpinski, W. (1988-02-15), "Chapter V.7", in Ed. A. Schinzel, Elementary Theory of Numbers, North-Holland Mathematical Library (2 Sub ed.), Amsterdam: North Holland, p. 232, ISBN 9780444866622

[lpsp-8] Robert Baillie; Samuel S. Wagstaff Jr. (October 1980). "Lucas Pseudoprimes" (PDF). Mathematics of Computation. 35 (152): 1391–1417. doi:10.1090/S0025-5718-1980-0583518-6. MR 0583518.

[9] "Pseudoprimzahlen: Tabelle Pseudoprimzahlen (15 - 4999) – Wikibooks, Sammlung freier Lehr-, Sach- und Fachbücher". de.m.wikibooks.org. Retrieved 21 April 2018.

[10] Michon, Gerard. "Pseudo-primes, Weak Pseudoprimes, Strong Pseudoprimes, Primality - Numericana". www.numericana.com. Retrieved 21 April 2018.

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