Fermat pseudoprime

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

Definition

Fermat's little theorem states that if p is prime and a is coprime to p, then ap−1 − 1 is divisible by p. For an integer a > 1, if a composite integer x divides ax−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 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.

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 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 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

Some sources use variations of the definition, for example to only allow odd numbers to be pseudoprimes.[2]

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:[3]

A composite number q is a Fermat pseudoprime to a base a, if a^{q-1} \equiv 1 \pmod q and 2 \le a \le q-2.

Properties

Distribution

There are infinitely many pseudoprimes to a given base (in fact, infinitely many strong pseudoprimes (see Theorem 1 of [4]) and infinitely many Carmichael numbers [5]) , 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 [4]).

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

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

(sequence A001567 in OEIS)

Poulet 1 to 15
34111 · 31
5613 · 11 · 17
6453 · 5 · 43
11055 · 13 · 17
138719 · 73
17297 · 13 · 19
19053 · 5 · 127
204723 · 89
24655 · 17 · 29
270137 · 73
28217 · 13 · 31
327729 · 113
403337 · 109
436917 · 257
43713 · 31 · 47
Poulet 16 to 30
468131 · 151
546143 · 127
66017 · 23 · 41
795773 · 109
832153 · 157
84813 · 11 · 257
89117 · 19 · 67
1026131 · 331
105855 · 29 · 73
113055 · 7 · 17 · 19
128013 · 17 · 251
137417 · 13 · 151
1374759 · 233
1398111 · 31 · 41
1449143 · 337
Poulet 31 to 45
1570923 · 683
158417 · 31 · 73
167055 · 13 · 257
187053 · 5 · 29 · 43
1872197 · 193
1995171 · 281
230013 · 11 · 17 · 41
2337797 · 241
257613 · 31 · 277
2934113 · 37 · 61
301217 · 13 · 331
3088917 · 23 · 79
3141789 · 353
3160973 · 433
31621103 · 307
Poulet 46 to 60
331533 · 43 · 257
349455 · 29 · 241
3533389 · 397
398655 · 7 · 17 · 67
410417 · 11 · 13 · 41
416655 · 13 · 641
42799127 · 337
4665713 · 37 · 97
49141157 · 313
49981151 · 331
526337 · 73 · 103
552453 · 5 · 29 · 127
574217 · 13 · 631
60701101 · 601
6078789 · 683

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

Smallest Fermat pseudoprimes

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  A090086, and it is 4, 341, 91, 15, 4, 35, 6, 9, 4, 9, 10, 65, 4, 15, 14, 15, 4, 25, 6, 21, 4, 21, 22, 25, 4, 9, 26, 9, 4, 49, 6, 25, 4, 15, 9, 35, 4, 39, 38, 39, 4, 205, 6, 9, 4, 9, 46, 49, 4, 21, 10, 51, 4, 55, 6, 15, 4, 57, 15, 341, 4, 9, 62, 9, 4, 65, 6, 25, 4, 69, 9, 85, 4, 15, 74, 15, 4, 77, 6, 9, 4, 9, 21, 85, 4, 15, 86, 87, 4, 91, 6, 21, 4, 15, 91, 65, 4, 9, 14, 9, 4, 133, 6, 15, 4, 15, 9, 91, 4, 111, 10, 65, 4, 91, 6, 9, 4, 9, 15, 77, 4, 33, 85, 15, 4, 25, 6, 49, ...)

(sequence A007535 in 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 · 13 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

First few Fermat pseudoprimes in base a (up to 10000)

a First few Fermat pseudoprimes in base a (up to 10000) 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 composite numbers) 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  A020159 to  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?

The following is a list about all base b < n which n is a Fermat pseudoprime (all composite number is a pseudoprime to base 1, and for b > n, the solutions are just shifted by k*n for k > 0), if a composite number n is not listed in the list (or n is in the sequence A209211 = 1, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 54, 56, 58, 60, 62, 64, 68, 72, 74, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 114, 116, 118, 120, 122, 126, 128, 132, 134, 136, 138, 140, 142, 144, 146, 150, ... ), then n is a pseudoprime only in base 1, or the bases which congruent to 1 (mod n) (that is, the number of the values of b is 1), these ns are up to 150)

n b which n is a Fermat pseudoprime (< n) number of the values of b (< n)
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

For more information (n = 151 to 5000), see table of pseudoprimes (text in German), 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 that 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 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 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, ... (sequence A064234 in OEIS) (if and only if n is a nontotient, then the nth term of this sequence is 0)

Weak pseudoprimes

A composite number n which satisfy that bn = b (mod n) is called weak pseudoprime to base b, the least weak pseudoprime to base b 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 OEIS)

If we require that n > b, they are

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 OEIS)

Euler–Jacobi pseudoprimes

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.

Applications

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

  1. ↑ 1.0 1.1 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.
  2. ↑ Weisstein, Eric W., "Fermat Pseudoprime", MathWorld.
  3. ↑ Richard Crandall, Carl Pomerance (2001). "Theorem 3.4.2". Prime Numbers – A Computational Perspective. Springer-Verlag. p. 132.
  4. ↑ 4.0 4.1 Carl Pomerance; John L. Selfridge, Samuel S. Wagstaff, Jr. (July 1980). "The pseudoprimes to 25·109" (PDF). Mathematics of Computation 35 (151): 1003–1026. doi:10.1090/S0025-5718-1980-0572872-7.
  5. ↑ W. R. Alford; Andrew Granville, Carl Pomerance (1994). "There are Infinitely Many Carmichael Numbers" (PDF). Annals of Mathematics 139: 703–722. doi:10.2307/2118576.

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