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In [[number theory]], the '''Fermat pseudoprimes''' make up the most important class of [[pseudoprime]]s that come from [[Fermat's little theorem]].
Irwin Butts is what my wife enjoys to contact me although I don't truly like being known as like that. Hiring is her working day job now but she's always wanted her own business. To gather coins is what her family members and her appreciate. Her husband and her reside in Puerto Rico but she will have to move one day or another.<br><br>Here is my homepage [http://rz.tl/healthymealsdelivered68340 rz.tl]
 
== Definition ==
[[Fermat's little theorem]] states that if ''p'' is prime and ''a'' is [[coprime]] to ''p'', then ''a''<sup>''p''−1</sup>&nbsp;−&nbsp;1 is [[Divisor|divisible]] by ''p''. If a composite integer ''x'' is coprime to an integer ''a'' > 1 and ''x'' divides ''a''<sup>''x''−1</sup>&nbsp;−&nbsp;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''.<ref name="desmedt-10-23">{{cite book|author=Desmedt, Yvo|chapter=Encryption Schemes|editors=[[Mikhail Atallah|Atallah, Mikhail J.]] & Blanton, Marina|title=Algorithms and theory of computation handbook: Special topics and techniques|publisher=CRC Press|year=2010|isbn=978-1-58488-820-8|pages=10–23|url=http://books.google.com/books?id=SbPpg_4ZRGsC&pg=SA10-PA23}}</ref>
 
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<sup>340</sup> ≡ 1 (mod 341) and thus passes
[[Fermat primality test]] for the base 2.
 
Pseudoprimes to base 2 are sometimes called '''Poulet numbers''', after the [[Belgium|Belgian]] mathematician [[Paul Poulet]], '''Sarrus 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]].<ref name="desmedt-10-23" />
 
=== Variations ===
Some sources use variations of the definition, for example to only allow odd numbers to be pseudoprimes.<ref>{{MathWorld|title=Fermat Pseudoprime|urlname=FermatPseudoprime}}</ref>
 
Every odd number ''q'' satisfies <math>a^{q-1} \equiv 1 \pmod q</math> for <math>a=q-1</math>. This trivial case is excluded in the definition of a Fermat pseudoprime given by [[Richard Crandall|Crandall]] and [[Carl Pomerance|Pomerance]]:<ref>{{cite book |author=Richard Crandall, Carl Pomerance |title=Prime Numbers – A Computational Perspective |publisher=Springer-Verlag |year=2001 |page=132 |chapter=Theorem 3.4.2}}</ref>
:A composite number ''q'' is a Fermat pseudoprime to a base ''a'', if <math>a^{q-1} \equiv 1 \pmod q</math> and <math>2 \le a \le q-2.</math>
 
==Properties==
===Distribution===
There are infinitely many pseudoprimes to a given base (in fact, infinitely many [[strong pseudoprime]]s (see Theorem 1 of
<ref name="PSW">{{cite journal|coauthors=[[John L. Selfridge]], [[Samuel S. Wagstaff, Jr.]]|title=The pseudoprimes to 25·10<sup>9</sup>|journal=Mathematics of Computation|date=July 1980|volume=35|issue=151|pages=1003–1026|url=http://www.math.dartmouth.edu/~carlp/PDF/paper25.pdf|author = [[Carl Pomerance]]| doi=10.1090/S0025-5718-1980-0572872-7 }}</ref>)
and infinitely many Carmichael numbers
<ref name="Alford1994">{{cite journal |author=[[W. R. (Red) Alford|W. R. Alford]] |coauthors=[[Andrew Granville]], [[Carl Pomerance]] |title=There are Infinitely Many Carmichael Numbers |journal=[[Annals of Mathematics]] |volume=139 |year=1994 |pages=703–722 |doi=10.2307/2118576 |url=http://www.math.dartmouth.edu/~carlp/PDF/paper95.pdf}}</ref>)
, but they are rather rare.
There are only three pseudoprimes to base 2 below 1000, 245 below one million, and only 21853 less than 25·10<sup>9</sup> (see Table 1 of <ref name="PSW"/>).
 
Starting at 17·257, the product of consecutive Fermat numbers is a base-2 pseudoprime.
 
===Factorizations===
The factorizations of the 60 Poulet numbers up to 60787, including 13 Carmichael numbers (in bold), are in the below 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 2<sup>''d''</sup> − 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.
 
{| class="wikitable"
|-
! ''a''
! smallest p-p
! ''a''
! smallest p-p
! ''a''
! smallest p-p
! ''a''
! smallest p-p
|-
| &nbsp;
| &nbsp;
| 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 · 13
| bgcolor="#FFEBAD" | 65
| bgcolor="#FFEBAD" | 112 = 2<sup>4</sup> · 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 = 3<sup>4</sup>
| 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
|}
 
== Euler–Jacobi pseudoprimes ==
{{main|Euler–Jacobi pseudoprime}}
Another approach is to use more refined notions of pseudoprimality, e.g. [[strong pseudoprime]]s or [[Euler–Jacobi pseudoprime]]s, for which there are no analogues of [[Carmichael number]]s. This leads to [[randomized algorithm|probabilistic algorithm]]s 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 (algorithm)|RSA]] require the ability to quickly find large primes. The usual algorithm to generate prime numbers is to generate random odd numbers and [[primality test|test]] them for primality. However, [[deterministic algorithm|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, [http://www.cecm.sfu.ca/Pseudoprimes/ Database of Base-2 Pseudoprimes up to 10^15] (including Strong Pseudoprimes and Carmichael numbers)
 
{{Classes of natural numbers}}
[[Category:Pseudoprimes]]
[[Category:Asymmetric-key algorithms]]

Latest revision as of 06:38, 6 June 2014

Irwin Butts is what my wife enjoys to contact me although I don't truly like being known as like that. Hiring is her working day job now but she's always wanted her own business. To gather coins is what her family members and her appreciate. Her husband and her reside in Puerto Rico but she will have to move one day or another.

Here is my homepage rz.tl

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