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. 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''n First few Fermat pseudoprimes in base n OEIS sequence1 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,... 2 341, 561, 645, 1105, 1387, 1729, 1905, 2047, 2465, 2701, 2821, 3277, 4033, 4369, 4371, 4681, 5461, 6601, 7957, 8321, 8481, 8911,... 3 91, 121, 286, 671, 703, 949, 1105, 1541, 1729, 1891, 2465, 2665, 2701, 2821, 3281, 3367, 3751, 4961, 5551, 6601, 7381, 8401, 8911,... 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,... 5 4, 124, 217, 561, 781, 1541, 1729, 1891, 2821, 4123, 5461, 5611, 5662, 5731, 6601, 7449, 7813, 8029, 8911, 9881,... 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,... 7 6, 25, 325, 561, 703, 817, 1105, 1825, 2101, 2353, 2465, 3277, 4525, 4825, 6697, 8321,... 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,... 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,... 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,... 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,... 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,... 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,... 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,... 15 14, 341, 742, 946, 1477, 1541, 1687, 1729, 1891, 1921, 2821, 3133, 3277, 4187, 6541, 6601, 7471, 8701, 8911, 9073,... 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,... 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,... 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,... 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,... 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,... 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,... 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,... 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,... 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,... 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,... 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,... 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,... 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,... 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,... 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,...
For more information, see to, and for all bases up to 150, see , this page does not define n is a pseudoprime to a base congruent to 1 or -1Which bases ''b'' make ''n'' a Fermat pseudoprime? If composite is even, then is a Fermat pseudoprime to the trivial base. If composite is odd, then is a Fermat pseudoprime to the trivial bases. For any composite, the number of distinct bases modulo, for which is a Fermat pseudoprime base, is where are the distinct prime factors of. This includes the trivial bases. For example, for , this product is. For, the smallest such nontrivial base is. Every odd composite is a Fermat pseudoprime to at least two nontrivial bases modulo unless is a power of 3. 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, then n is a pseudoprime only to the trivial base 1 modulo n .n bases b to which n is a Fermat pseudoprime number of the bases of b 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, see, this page does not define n is a pseudoprime to a base congruent to 1 or -1. When p is a prime, p 2 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 The least base b > 1 which n is a pseudoprime to base b are The number of the values of b for n must divides Euler phi function|, or = 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 least number with n values of b areWeak pseudoprimes A composite number n which satisfy that is called weak pseudoprime to base b . A pseudoprime to base a 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. The least weak pseudoprime to base b = 1, 2,... are: 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 less than 561 occurs in the above sequences if and only if p − 1 divides q − 1. Besides, the smallest pseudoprime to base n is also usually semiprime, the first counterexample is = 385 = 5 × 7 × 11. If we require n > b , they are Carmichael numbers are weak pseudoprimes to all bases. The smallest even weak pseudoprime in base 2 is 161038.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", 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.
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