Editing Fermat pseudoprime (section)

==Properties==

===Distribution===
There are infinitely many pseudoprimes to any given base <math>a > 1</math>. In 1904, Cipolla showed how to produce an infinite number of pseudoprimes to base <math>a > 1</math>: let <math>A = (a^p-1)/(a-1)</math> and let <math>B = (a^p+1)/(a+1)</math>, where <math>p</math> is a prime number that does not divide <math>a(a^2-1)</math>. Then <math>n= AB</math> is composite, and is a pseudoprime to base <math>a</math>.<ref>{{ 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 }}</ref><ref>{{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}}</ref> 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 pseudoprime]]s to any base greater than 1 (see Theorem 1 of
<ref name="PSW">{{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 25·10<sup>9</sup> |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 }}</ref>) and infinitely many Carmichael numbers,<ref name="Alford1994">{{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 }}</ref> but they are comparatively rare. There are three pseudoprimes to base 2 below 1000, 245 below one million, and 21853 less than 25·10<sup>9</sup>. There are 4842 strong pseudoprimes base 2 and 2163 Carmichael numbers below this limit (see Table 1 of <ref name="PSW"/>).

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

The probability of a composite number n passing the Fermat test approaches zero for <math>n \to\infty</math>. Specifically, Kim and Pomerance showed the following: The probability that a random odd number  <math>n \le x</math> is a Fermat pseudoprime to a random base <math>1<b<n-1</math> is less than 2.77·10<sup>−8</sup> for x= 10<sup>100</sup>, and is at most (log x)<sup>−197</sup><10<sup>−10,000</sup> for x≥10<sup>100,000</sup>.<ref>{{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 }}</ref>

===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
|-
|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.<ref>{{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 }}</ref>

=== 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'')