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 <math>p</math> is prime and <math>a</math> is coprime to <math>p</math>, then <math>a^{p-1}-1</math> is divisible by <math>p</math>. For a positive integer <math>a</math>, if a composite integer <math>x</math> divides <math>a^{x-1}-1</math> then <math>x</math> is called a Fermat pseudoprime to base <math>a</math>.

In other words, a composite integer is a Fermat pseudoprime to base <math>a</math> if it successfully passes the Fermat primality test for the base <math>a</math>. The false statement that all numbers that pass the Fermat primality test for base 2 are prime is called the Chinese hypothesis.

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: <math>2^{340} \equiv 1 (\bmod{341})</math> and thus passes the

Fermat primality test for the base 2.

Pseudoprimes to base 2 are sometimes called Sarrus numbers, after P. F. Sarrus who discovered that 341 has this property, Poulet numbers, after P. Poulet who made a table of such numbers, or Fermatians .

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

An integer <math>x</math> that is a Fermat pseudoprime for all values of <math>a</math> that are coprime to <math>x</math> is called a Carmichael number. For example, if <math>a=2</math> and <math>p=5</math>, then <math>A=31</math>, <math>B=11</math>, and <math>n=AB=341</math> is a pseudoprime to base <math>2</math>.

In fact, there are infinitely many strong pseudoprimes to any base greater than 1 (see Theorem 1 of

) and infinitely many Carmichael numbers, but they are comparatively rare. There are three pseudoprimes to base 2 below 1000, 245 below one million, and 21853 less than 25·109. There are 4842 strong pseudoprimes base 2 and 2163 Carmichael numbers below this limit (see Table 1 of For a fixed base b, the proportion of pseudoprimes up to x is at most e-log x log log log x/(2 log log x) for large x.

Factorizations

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

{| border="0" cellpadding="0" cellspacing="0"

|- valign="top"

{| class="wikitable"

|+ 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

{| class="wikitable"

|+ 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

{| class="wikitable"

|+ 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

{| class="wikitable"

|+ 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 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 )

{| class="wikitable"

! a

! smallest p-p

! a

! smallest p-p

! a

! smallest p-p

! a

! smallest p-p

| bgcolor="#FFCBCB" | 1

| bgcolor="#FFCBCB" | 4 = 2²

| 51

| 65 = 5 · 13

| bgcolor="#FFEBAD" | 101

| bgcolor="#FFEBAD" | 175 = 5² · 7

| bgcolor="#FFEBAD" | 151

| bgcolor="#FFEBAD" | 175 = 5² · 7

| 2

| 341 = 11 · 31

| 52

| 85 = 5 · 17

| 102

| 133 = 7 · 19

| bgcolor="#FFEBAD" | 152

| bgcolor="#FFEBAD" | 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

| bgcolor="#B3B7FF" | 104

| bgcolor="#B3B7FF" | 105 = 3 · 5 · 7

| 154

| 155 = 5 · 31

| bgcolor="#FFEBAD" | 5

| bgcolor="#FFEBAD" | 124 = 2² · 31

| bgcolor="#FFEBAD" | 55

| bgcolor="#FFEBAD" | 63 = 3² · 7

| 105

| 451 = 11 · 41

| bgcolor="#B3B7FF" | 155

| bgcolor="#B3B7FF" | 231 = 3 · 7 · 11

| 6

| 35 = 5 · 7

| 56

| 57 = 3 · 19

| 106

| 133 = 7 · 19

| 156

| 217 = 7 · 31

| bgcolor="#FFCBCB" | 7

| bgcolor="#FFCBCB" | 25 = 5²

| 57

| 65 = 5 · 13

| 107

| 133 = 7 · 19

| bgcolor="#B3B7FF" | 157

| bgcolor="#B3B7FF" | 186 = 2 · 3 · 31

| bgcolor="#FFCBCB" | 8

| bgcolor="#FFCBCB" | 9 = 3²

| 58

| 133 = 7 · 19

| 108

| 341 = 11 · 31

| 158

| 159 = 3 · 53

| bgcolor="#FFEBAD" | 9

| bgcolor="#FFEBAD" | 28 = 2² · 7

| 59

| 87 = 3 · 29

| bgcolor="#FFEBAD" | 109

| bgcolor="#FFEBAD" | 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

| bgcolor="#B3B7FF" | 111

| bgcolor="#B3B7FF" | 190 = 2 · 5 · 19

| bgcolor="#B3B7FF" | 161

| bgcolor="#B3B7FF" | 190 = 2 · 5 · 19

| 12

| 65 = 5 · 13

| bgcolor="#FFEBAD" | 62

| bgcolor="#FFEBAD" | 63 = 3² · 7

| bgcolor="#FFCBCB" | 112

| bgcolor="#FFCBCB" | 121 = 11²

| 162

| 481 = 13 · 37

| 13

| 21 = 3 · 7

| 63

| 341 = 11 · 31

| 113

| 133 = 7 · 19

| bgcolor="#B3B7FF" | 163

| bgcolor="#B3B7FF" | 186 = 2 · 3 · 31

| 14

| 15 = 3 · 5

| 64

| 65 = 5 · 13

| 114

| 115 = 5 · 23

| bgcolor="#B3B7FF" | 164

| bgcolor="#B3B7FF" | 165 = 3 · 5 · 11

| 15

| 341 = 11 · 31

| bgcolor="#FFEBAD" | 65

| bgcolor="#FFEBAD" | 112 = 2⁴ · 7

| 115

| 133 = 7 · 19

| bgcolor="#FFEBAD" | 165

| bgcolor="#FFEBAD" | 172 = 2² · 43

| 16

| 51 = 3 · 17

| 66

| 91 = 7 · 13

| bgcolor="#FFEBAD" | 116

| bgcolor="#FFEBAD" | 117 = 3² · 13

| 166

| 301 = 7 · 43

| bgcolor="#FFEBAD" | 17

| bgcolor="#FFEBAD" | 45 = 3² · 5

| 67

| 85 = 5 · 17

| 117

| 145 = 5 · 29

| bgcolor="#B3B7FF" | 167

| bgcolor="#B3B7FF" | 231 = 3 · 7 · 11

| bgcolor="#FFCBCB" | 18

| bgcolor="#FFCBCB" | 25 = 5²

| 68

| 69 = 3 · 23

| 118

| 119 = 7 · 17

| bgcolor="#FFCBCB" | 168

| bgcolor="#FFCBCB" | 169 = 13²

| bgcolor="#FFEBAD" | 19

| bgcolor="#FFEBAD" | 45 = 3² · 5

| 69

| 85 = 5 · 17

| 119

| 177 = 3 · 59

| bgcolor="#B3B7FF" | 169

| bgcolor="#B3B7FF" | 231 = 3 · 7 · 11

| 20

| 21 = 3 · 7

| bgcolor="#FFCBCB" | 70

| bgcolor="#FFCBCB" | 169 = 13²

| bgcolor="#FFCBCB" | 120

| bgcolor="#FFCBCB" | 121 = 11²

| bgcolor="#FFEBAD" | 170

| bgcolor="#FFEBAD" | 171 = 3² · 19

| 21

| 55 = 5 · 11

| bgcolor="#B3B7FF" | 71

| bgcolor="#B3B7FF" | 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

| bgcolor="#FFCBCB" | 24

| bgcolor="#FFCBCB" | 25 = 5²

| bgcolor="#FFEBAD" | 74

| bgcolor="#FFEBAD" | 75 = 3 · 5²

| bgcolor="#FFEBAD" | 124

| bgcolor="#FFEBAD" | 125 = 5³

| bgcolor="#FFEBAD" | 174

| bgcolor="#FFEBAD" | 175 = 5² · 7

| bgcolor="#FFEBAD" | 25

| bgcolor="#FFEBAD" | 28 = 2² · 7

| 75

| 91 = 7 · 13

| 125

| 133 = 7 · 19

| 175

| 319 = 11 · 19

| bgcolor="#FFEBAD" | 26

| bgcolor="#FFEBAD" | 27 = 3³

| 76

| 77 = 7 · 11

| 126

| 247 = 13 · 19

| 176

| 177 = 3 · 59

| 27

| 65 = 5 · 13

| 77

| 247 = 13 · 19

| bgcolor="#FFEBAD" | 127

| bgcolor="#FFEBAD" | 153 = 3² · 17

| bgcolor="#FFEBAD" | 177

| bgcolor="#FFEBAD" | 196 = 2² · 7²

| bgcolor="#FFEBAD" | 28

| bgcolor="#FFEBAD" | 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

| bgcolor="#FFCBCB" | 30

| bgcolor="#FFCBCB" | 49 = 7²

| bgcolor="#FFEBAD" | 80

| bgcolor="#FFEBAD" | 81 = 3⁴

| 130

| 217 = 7 · 31

| 180

| 217 = 7 · 31

| bgcolor="#FFCBCB" | 31

| bgcolor="#FFCBCB" | 49 = 7²

| 81

| 85 = 5 · 17

| 131

| 143 = 11 · 13

| bgcolor="#B3B7FF" | 181

| bgcolor="#B3B7FF" | 195 = 3 · 5 · 13

| 32

| 33 = 3 · 11

| 82

| 91 = 7 · 13

| 132

| 133 = 7 · 19

| 182

| 183 = 3 · 61

| 33

| 85 = 5 · 17

| bgcolor="#B3B7FF" | 83

| bgcolor="#B3B7FF" | 105 = 3 · 5 · 7

| 133

| 145 = 5 · 29

| 183

| 221 = 13 · 17

| 34

| 35 = 5 · 7

| 84

| 85 = 5 · 17

| bgcolor="#FFEBAD" | 134

| bgcolor="#FFEBAD" | 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

| bgcolor="#FFEBAD" | 37

| bgcolor="#FFEBAD" | 45 = 3² · 5

| 87

| 91 = 7 · 13

| bgcolor="#FFEBAD" | 137

| bgcolor="#FFEBAD" | 148 = 2² · 37

| 187

| 217 = 7 · 31

| 38

| 39 = 3 · 13

| 88

| 91 = 7 · 13

| 138

| 259 = 7 · 37

| bgcolor="#FFEBAD" | 188

| bgcolor="#FFEBAD" | 189 = 3³ · 7

| 39

| 95 = 5 · 19

| bgcolor="#FFEBAD" | 89

| bgcolor="#FFEBAD" | 99 = 3² · 11

| 139

| 161 = 7 · 23

| 189

| 235 = 5 · 47

| 40

| 91 = 7 · 13

| 90

| 91 = 7 · 13

| 140

| 141 = 3 · 47

| bgcolor="#B3B7FF" | 190

| bgcolor="#B3B7FF" | 231 = 3 · 7 · 11

| bgcolor="#B3B7FF" | 41

| bgcolor="#B3B7FF" | 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

| bgcolor="#FFEBAD" | 193

| bgcolor="#FFEBAD" | 276 = 2² · 3 · 23

| bgcolor="#FFEBAD" | 44

| bgcolor="#FFEBAD" | 45 = 3² · 5

| 94

| 95 = 5 · 19

| 144

| 145 = 5 · 29

| bgcolor="#B3B7FF" | 194

| bgcolor="#B3B7FF" | 195 = 3 · 5 · 13

| bgcolor="#FFEBAD" | 45

| bgcolor="#FFEBAD" | 76 = 2² · 19

| 95

| 141 = 3 · 47

| bgcolor="#FFEBAD" | 145

| bgcolor="#FFEBAD" | 153 = 3² · 17

| 195

| 259 = 7 · 37

| 46

| 133 = 7 · 19

| 96

| 133 = 7 · 19

| bgcolor="#FFEBAD" | 146

| bgcolor="#FFEBAD" | 147 = 3 · 7²

| 196

| 205 = 5 · 41

| 47

| 65 = 5 · 13

| bgcolor="#B3B7FF" | 97

| bgcolor="#B3B7FF" | 105 = 3 · 5 · 7

| bgcolor="#FFCBCB" | 147

| bgcolor="#FFCBCB" | 169 = 13²

| bgcolor="#B3B7FF" | 197

| bgcolor="#B3B7FF" | 231 = 3 · 7 · 11

| bgcolor="#FFCBCB" | 48

| bgcolor="#FFCBCB" | 49 = 7²

| bgcolor="#FFEBAD" | 98

| bgcolor="#FFEBAD" | 99 = 3² · 11

| bgcolor="#B3B7FF" | 148

| bgcolor="#B3B7FF" | 231 = 3 · 7 · 11

| 198

| 247 = 13 · 19

| bgcolor="#B3B7FF" | 49

| bgcolor="#B3B7FF" | 66 = 2 · 3 · 11

| 99

| 145 = 5 · 29

| bgcolor="#FFEBAD" | 149

| bgcolor="#FFEBAD" | 175 = 5² · 7

| bgcolor="#FFEBAD" | 199

| bgcolor="#FFEBAD" | 225 = 3² · 5²

| 50

| 51 = 3 · 17

| bgcolor="#FFEBAD" | 100

| bgcolor="#FFEBAD" | 153 = 3² · 17

| bgcolor="#FFCBCB" | 150

| bgcolor="#FFCBCB" | 169 = 13²

| 200

| 201 = 3 · 67

List of Fermat pseudoprimes in fixed base n

{|class="wikitable"

|First few Fermat pseudoprimes in base n

|OEIS sequence

|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)

|341, 561, 645, 1105, 1387, 1729, 1905, 2047, 2465, 2701, 2821, 3277, 4033, 4369, 4371, 4681, 5461, 6601, 7957, 8321, 8481, 8911, ...

|91, 121, 286, 671, 703, 949, 1105, 1541, 1729, 1891, 2465, 2665, 2701, 2821, 3281, 3367, 3751, 4961, 5551, 6601, 7381, 8401, 8911, ...

|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, ...

|4, 124, 217, 561, 781, 1541, 1729, 1891, 2821, 4123, 5461, 5611, 5662, 5731, 6601, 7449, 7813, 8029, 8911, 9881, ...

|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, ...

|6, 25, 325, 561, 703, 817, 1105, 1825, 2101, 2353, 2465, 3277, 4525, 4825, 6697, 8321, ...

|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, ...

|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 (base 31 to 100), see to , 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)

Bases b for which n is a Fermat pseudoprime

If composite <math>n</math> is even, then <math>n</math> is a Fermat pseudoprime to the trivial base <math>b \equiv 1 \pmod n</math>.

If composite <math>n</math> is odd, then <math>n</math> is a Fermat pseudoprime to the trivial bases <math>b \equiv \pm 1 \pmod n</math>.

For any composite <math>n</math>, the number of distinct bases <math>b</math> modulo <math>n</math>, for which <math>n</math> is a Fermat pseudoprime base <math>b</math>, is

:<math> \prod_{i=1}^k \gcd(p_i - 1, n - 1)</math>

where <math>p_1, \dots, p_k</math> are the distinct prime factors of <math>n</math>. This includes the trivial bases.

For example, for <math>n = 341 = 11 \cdot 31</math>, this product is <math>\gcd(10, 340) \cdot \gcd(30, 340) = 100</math>. For <math>n = 341</math>, the smallest such nontrivial base is <math>b = 2</math>.

Every odd composite <math>n</math> is a Fermat pseudoprime to at least two nontrivial bases modulo <math>n</math> unless <math>n</math> is a power of 3. For example, the smallest even weak pseudoprime to base 2 is 161038 (see ).

The least weak pseudoprime to bases 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, ...

Carmichael numbers are weak pseudoprimes to all bases, thus all terms in this list are less than or equal to the smallest Carmichael number, 561. Except for 561 = 3⋅11⋅17, only semiprimes can occur in the above sequence. 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 ). The least Fermat pseudoprime to base b (also not necessary exceeding b) () is usually semiprime, but not always; the first counterexample is (648) = 385 = 5 × 7 × 11.

If we require n > b, the least weak pseudoprimes (for b = 1, 2, ...) 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, ...

Euler pseudoprimes

Euler pseudoprimes are a subset of the Fermat pseudoprimes for any given base <math>a</math>. There also exists absolute Euler pseudoprimes, which are a subset of the Carmichael numbers, and which are the analogue of Carmichael numbers for Euler pseudoprimes.

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

External links

W. F. Galway and Jan Feitsma, Tables of pseudoprimes to base 2 and related data (comprehensive list of all pseudoprimes to base 2 below 264, including factorization, strong pseudoprimes, and Carmichael numbers)
A research for pseudoprime