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Fermat pseudoprime

{{Short description|Composite number that passes Fermat's probable primality test}}

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 a^{p-1}-1 is divisible by p. For a positive integer a, if a composite integer x divides a^{x-1}-1 then x is called a Fermat pseudoprime to base a. {{cite book | author = Samuel S. Wagstaff Jr. |author-link = Samuel S. Wagstaff, Jr. | title = The Joy of Factoring | publisher =American Mathematical Society | location = Providence, RI | year = 2013 | isbn = 978-1-4704-1048-3 |url =https://www.ams.org/bookpages/stml-68 }}{{rp|Def. 3.32}}

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.{{cite book |last=Desmedt |first=Yvo |editor1-last=Atallah |editor1-first=Mikhail J. |editor1-link=Mikhail Atallah |editor2-last=Blanton |editor2-first=Marina |year=2010 |title=Algorithms and theory of computation handbook: Special topics and techniques |chapter=Encryption Schemes |publisher=CRC Press |isbn=978-1-58488-820-8 |pages=10–23 |chapter-url=https://books.google.com/books?id=SbPpg_4ZRGsC&pg=SA10-PA23}} 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: 2^{340} \equiv 1 (\bmod{341}) 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 {{OEIS|id=A001567}}.

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.{{rp|Def. 3.34}}

Properties

=Distribution=

There are infinitely many pseudoprimes to any given base a > 1. In 1904, Cipolla showed how to produce an infinite number of pseudoprimes to base a > 1: let A = (a^p-1)/(a-1) and let B = (a^p+1)/(a+1), where p is a prime number that does not divide a(a^2-1). Then n= AB is composite, and is a pseudoprime to base a.{{ cite book | page=108 | author=Paulo Ribenboim |author-link=Paulo Ribenboim | title=The New Book of Prime Number Records | isbn=0-387-94457-5 | publisher=Springer-Verlag | location=New York | year=1996 }}{{cite journal | url=https://cs.uwaterloo.ca/journals/JIS/VOL10/Hamahata2/hamahata44.pdf | title=Cipolla Pseudoprimes | last1=Hamahata | first1=Yoshinori | last2=Kokubun | first2=Y. | journal=Journal of Integer Sequences | date=2007 | volume=10| issue=8}} For example, if a=2 and p=5, then A=31, B=11, and n=AB=341 is a pseudoprime to base 2.

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

{{cite journal |last1=Pomerance |first1=Carl |author-link1=Carl Pomerance |last2=Selfridge |first2=John L. |author-link2=John L. Selfridge |last3=Wagstaff |first3=Samuel S. Jr. |author-link3=Samuel S. Wagstaff Jr. |date=July 1980 |title=The pseudoprimes to 25·109 |journal=Mathematics of Computation |doi=10.1090/S0025-5718-1980-0572872-7 |volume=35 |issue=151 |pages=1003–1026 |url=http://www.math.dartmouth.edu/~carlp/PDF/paper25.pdf |archive-url=https://web.archive.org/web/20050304202721/http://math.dartmouth.edu/~carlp/PDF/paper25.pdf |archive-date=2005-03-04 |url-status=live |doi-access=free }}) and infinitely many Carmichael numbers,{{cite journal |last1=Alford |first1=W. R. |author-link=W. R. (Red) Alford |last2=Granville |first2=Andrew |author-link2=Andrew Granville |last3=Pomerance |first3=Carl |author-link3=Carl Pomerance |year=1994 |title=There are Infinitely Many Carmichael Numbers |journal=Annals of Mathematics |doi=10.2307/2118576 |volume=140 |issue=3 |pages=703–722 |url=http://www.math.dartmouth.edu/~carlp/PDF/paper95.pdf |archive-url=https://web.archive.org/web/20050304203448/http://math.dartmouth.edu/~carlp/PDF/paper95.pdf |archive-date=2005-03-04 |url-status=live |jstor=2118576 }} 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 ).

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

The probability of a composite number n passing the Fermat test approaches zero for n \to\infty. Specifically, Kim and Pomerance showed the following: The probability that a random odd number n \le x is a Fermat pseudoprime to a random base 1 is less than 2.77·10−8 for x= 10100, and is at most (log x)−197<10−10,000 for x≥10100,000.{{cite journal | url=https://www.jstor.org/stable/2008733 | jstor=2008733 | title=The Probability that a Random Probable Prime is Composite | last1=Kim | first1=Su Hee | last2=Pomerance | first2=Carl | journal=Mathematics of Computation | date=1989 | volume=53 | issue=188 | pages=721–741 | doi=10.2307/2008733 }}

=Factorizations=

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

{{OEIS|id=A001567}}

border="0" cellpadding="0" cellspacing="0"
valign="top"

|

{| class="wikitable"

|+ 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

|

class="wikitable"

|+ 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

|

class="wikitable"

|+ 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

|

class="wikitable"

|+ 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.{{Citation

|date = 1988-02-15

|edition = 2 Sub

|isbn = 9780444866622

|location = Amsterdam

|publisher = North Holland

|title = Elementary Theory of Numbers

|first = W.

|last = Sierpinski

|chapter = Chapter V.7

|page = 232

|series = North-Holland Mathematical Library

|editor = Ed. A. Schinzel }}

= 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 {{oeis|id=A090086}})

{{OEIS|id=A007535}}

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"

|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)

|{{OEIS link|id=A002808}}

2

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

|{{OEIS link|id=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, ...

|{{OEIS link|id=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, ...

|{{OEIS link|id=A020136}}

5

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

|{{OEIS link|id=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, ...

|{{OEIS link|id=A005937}}

7

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

|{{OEIS link|id=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, ...

|{{OEIS link|id=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, ...

|{{OEIS link|id=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, ...

|{{OEIS link|id=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, ...

|{{OEIS link|id=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, ...

|{{OEIS link|id=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, ...

|{{OEIS link|id=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, ...

|{{OEIS link|id=A020142}}

15

|14, 341, 742, 946, 1477, 1541, 1687, 1729, 1891, 1921, 2821, 3133, 3277, 4187, 6541, 6601, 7471, 8701, 8911, 9073, ...

|{{OEIS link|id=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, ...

|{{OEIS link|id=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, ...

|{{OEIS link|id=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, ...

|{{OEIS link|id=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, ...

|{{OEIS link|id=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, ...

|{{OEIS link|id=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, ...

|{{OEIS link|id=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, ...

|{{OEIS link|id=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, ...

|{{OEIS link|id=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, ...

|{{OEIS link|id=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, ...

|{{OEIS link|id=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, ...

|{{OEIS link|id=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, ...

|{{OEIS link|id=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, ...

|{{OEIS link|id=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, ...

|{{OEIS link|id=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, ...

|{{OEIS link|id=A020158}}

For more information (base 31 to 100), see {{oeis|id=A020159}} to {{oeis|id=A020228}}, and for all bases up to 150, see [http://de.m.wikibooks.org/wiki/Pseudoprimzahlen:_Tabelle_Fermatsche_Pseudoprimzahlen 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 n is even, then n is a Fermat pseudoprime to the trivial base b \equiv 1 \pmod n.

If composite n is odd, then n is a Fermat pseudoprime to the trivial bases b \equiv \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

{{cite journal |author=Robert Baillie |author2=Samuel S. Wagstaff Jr. |author2-link=Samuel S. Wagstaff Jr. |title=Lucas Pseudoprimes|journal=Mathematics of Computation|date=October 1980|volume=35|issue=152|pages=1391–1417|url=http://mpqs.free.fr/LucasPseudoprimes.pdf |archive-url=https://web.archive.org/web/20060906205655/http://mpqs.free.fr/LucasPseudoprimes.pdf |archive-date=2006-09-06 |url-status=live| mr=583518| doi=10.1090/S0025-5718-1980-0583518-6 |doi-access=free}}{{rp|Thm. 1, p. 1392}}

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

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

For example, for n = 341 = 11 \cdot 31, this product is \gcd(10, 340) \cdot \gcd(30, 340) = 100. For 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.{{rp|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 {{OEIS link|id=A209211}}), then n is a pseudoprime only to the trivial base 1 modulo n.

class="wikitable"

|n

|bases 0 < b < n to which n is a Fermat pseudoprime

|# of bases
{{OEIS2C|id=A063994}}

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 {{lang|de|b:de:Pseudoprimzahlen: Tabelle Pseudoprimzahlen (15 - 4999)|italic=no}} (Table of pseudoprimes 16–4999). Unlike the list above, that page excludes the bases 1 and n−1. 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, ... {{OEIS|id=A063994}}

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, ... {{OEIS|id=A105222}}

The number of the values of b for a given n must divide \varphi(n), or {{OEIS link|id=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, ... {{OEIS link|id=A002997}}) The quotient = 2 if and only if n is in the sequence: 4, 6, 15, 91, 703, 1891, 2701, 11305, 12403, 13981, 18721, ... {{OEIS link|id=A191311}}.)

The least numbers which are pseudoprime to k 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, ... {{OEIS|id=A064234}} (if and only if k is even and not totient of squarefree number, then the kth term of this sequence is 0.)

Weak pseudoprimes

A composite number n which satisfies b^n \equiv b \pmod n is called a weak pseudoprime to base b. For any given base b, all Fermat pseudoprimes are weak pseudoprimes, and all weak pseudoprimes coprime to b are Fermat pseudoprimes. However, this definition also permits some pseudoprimes which are not coprime to b.{{cite web |url=http://www.numericana.com/answer/pseudo.htm#weak |title=Pseudo-primes, Weak Pseudoprimes, Strong Pseudoprimes, Primality |first=Gérard P. |last=Michon |website=Numericana |date=19 November 2003 |access-date=21 April 2018}} For example, the smallest even weak pseudoprime to base 2 is 161038 (see {{oeis|id=A006935}}).

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, ... {{OEIS2C|id=A000790}}

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 {{oeis|id=A108574}}). The least Fermat pseudoprime to base b (also not necessary exceeding b) ({{oeis|A090086}}) is usually semiprime, but not always; the first counterexample is {{OEIS link|A090086}}(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, ... {{OEIS2C|id=A239293}}

Euler–Jacobi pseudoprimes

{{main|Euler–Jacobi pseudoprime}}

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

{{Reflist}}

External links

  • W. F. Galway and Jan Feitsma, [http://www.cecm.sfu.ca/Pseudoprimes/ 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)
  • [http://www.numericana.com/answer/pseudo.htm A research for pseudoprime]

{{Classes of natural numbers}}

{{Pierre de Fermat}}

Category:Pseudoprimes

Category:Asymmetric-key algorithms