Fermat Pseudoprime

8/19/2019 Fermat Pseudoprime

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Fermat pseudoprimeFrom Wikipedia, the free encyclopedia

In number theory, the Fermat pseudoprimes make up the most important class of pseudoprimes that come from Fermat's little theorem.

Contents

1 Definition1.1 Variations

2 Properties2.1 Distribution2.2 Factorizations2.3 Smallest Fermat pseudoprimes2.4 First few Fermat pseudoprimes in base a (up to 10000)

3 Which bases b make n a Fermat pseudoprime?

4 Weak pseudoprimes5 Euler –Jacobi pseudoprimes6 Applications7 References8 External links

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 an integer a > 1, if a composite integer x divides

a x−1 − 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 the Fermat primality test for the base a.[1] It follows that if x is a Fermat pseudoprime to base a, then x is coprime to a.

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: 2340 ≡ 1 (mod 341) and

thus passes the Fermat primality test for the base 2.

https://en.wikipedia.org/wiki/Fermat_primality_testhttps://en.wikipedia.org/wiki/Fermat_primality_testhttps://en.wikipedia.org/wiki/Fermat_primality_testhttps://en.wikipedia.org/wiki/Divisorhttps://en.wikipedia.org/wiki/Coprimehttps://en.wikipedia.org/wiki/Fermat%27s_little_theoremhttps://en.wikipedia.org/wiki/Fermat%27s_little_theoremhttps://en.wikipedia.org/wiki/Pseudoprimehttps://en.wikipedia.org/wiki/Number_theory


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Pseudoprimes to base 2 are sometimes called Poulet numbers, after the Belgian mathematician Paul Poulet, Sarrus numbers, or Fermatians

(sequence A001567 in OEIS).

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.[1]

Variations

Some sources use variations of the definition, for example to only allow odd numbers to be pseudoprimes.[2]

Every odd number q satisfies for . This trivial case is excluded in the definition of a Fermat pseudoprime given by

Crandall and Pomerance:[3]

A composite number q is a Fermat pseudoprime to a base a, if and

Properties

Distribution

There are infinitely many pseudoprimes to a given base (in fact, infinitely many strong pseudoprimes (see Theorem 1 of [4]) and infinitely many

Carmichael numbers [5]) , but they are rather rare. There are only three pseudoprimes to base 2 below 1000, 245 below one million, and only 21853 less

than 25·109 (see Table 1 of [4]).

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

Factorizations

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

(sequence A001567 in OEIS)

https://en.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequenceshttps://oeis.org/A001567https://en.wikipedia.org/wiki/Mersenne_primehttps://en.wikipedia.org/wiki/Fermat_primehttps://en.wikipedia.org/wiki/Strong_pseudoprimehttps://en.wikipedia.org/wiki/Carl_Pomerancehttps://en.wikipedia.org/wiki/Richard_Crandallhttps://en.wikipedia.org/wiki/Carmichael_numberhttps://en.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequenceshttps://oeis.org/A001567https://en.wikipedia.org/wiki/Paul_Poulethttps://en.wikipedia.org/wiki/Belgium


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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 · 1094369 17 · 257

4371 3 · 31 · 47

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 · 23313981 11 · 31 · 41

14491 43 · 337

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 · 35331609 73 · 433

31621 103 · 307

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 · 63160701 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.[6]

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 A090086) (sequence A007535 in

OEIS)

https://en.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequenceshttps://oeis.org/A007535https://oeis.org/A090086https://en.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequenceshttps://en.wikipedia.org/wiki/Super-Poulet_number


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a smallest p-p a smallest p-p a smallest p-p a smallest p-p

1 4 = 2² 51 65 = 5 · 13 101 175 = 5² · 7 151 175 = 5² · 7

2 341 = 11 · 31 52 85 = 5 · 17 102 133 = 7 · 19 152 153 = 3² · 17

3 91 = 7 · 13 53 65 = 5 · 13 103 133 = 7 · 19 153 209 = 11 · 19

4 15 = 3 · 5 54 55 = 5 · 11 104 105 = 3 · 5 · 7 154 155 = 5 · 31

5 124 = 2² · 31 55 63 = 3² · 7 105 451 = 11 · 41 155 231 = 3 · 7 · 11

6 35 = 5 · 7 56 57 = 3 · 19 106 133 = 7 · 19 156 217 = 7 · 31

7 25 = 5² 57 65 = 5 · 13 107 133 = 7 · 19 157 186 = 2 · 3 · 31

8 9 = 3² 58 133 = 7 · 19 108 341 = 11 · 31 158 159 = 3 · 53

9 28 = 2² · 7 59 87 = 3 · 29 109 117 = 3² · 13 159 247 = 13 · 19

10 33 = 3 · 11 60 341 = 11 · 31 110 111 = 3 · 37 160 161 = 7 · 23

11 15 = 3 · 5 61 91 = 7 · 13 111 190 = 2 · 5 · 19 161 190=2 · 5 · 19

12 65 = 5 · 13 62 63 = 3² · 7 112 121 = 11² 162 481 = 13 · 37

13 21 = 3 · 7 63 341 = 11 · 31 113 133 = 7 · 19 163 186 = 2 · 3 · 31

14 15 = 3 · 5 64 65 = 5 · 13 114 115 = 5 · 23 164 165 = 3 · 5 · 11

15 341 = 11 · 13 65 112 = 2⁴ · 7 115 133 = 7 · 19 165 172 = 2² · 43

16 51 = 3 · 17 66 91 = 7 · 13 116 117 = 3² · 13 166 301 = 7 · 43

17 45 = 3² · 5 67 85 = 5 · 17 117 145 = 5 · 29 167 231 = 3 · 7 · 11

18 25 = 5² 68 69 = 3 · 23 118 119 = 7 · 17 168 169 = 13²

19 45 = 3² · 5 69 85 = 5 · 17 119 177 = 3 · 59 169 231 = 3 · 7 · 11

20 21 = 3 · 7 70 169 = 13² 120 121 = 11² 170 171 = 3² · 1921 55 = 5 · 11 71 105 = 3 · 5 · 7 121 133 = 7 · 19 171 215 = 5 · 43

22 69 = 3 · 23 72 85 = 5 · 17 122 123 = 3 · 41 172 247 = 13 · 19

23 33 = 3 · 11 73 111 = 3 · 37 123 217 = 7 · 31 173 205 = 5 · 41

24 25 = 5² 74 75 = 3 · 5² 124 125 = 5³ 174 175 = 5² · 7

25 28 = 2² · 7 75 91 = 7 · 13 125 133 = 7 · 19 175 319 = 11 · 19


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26 27 = 3³ 76 77 = 7 · 11 126 247 = 13 · 19 176 177 = 3 · 59

27 65 = 5 · 13 77 247 = 13 · 19 127 153 = 3² · 17 177 196 = 2² · 7²

28 45 = 3² · 5 78 341 = 11 · 31 128 129 = 3 · 43 178 247 = 13 · 19

29 35 = 5 · 7 79 91 = 7 · 13 129 217 = 7 · 31 179 185 = 5 · 37

30 49 = 7² 80 81 = 3⁴ 130 217 = 7 · 31 180 217 = 7 · 31

31 49 = 7² 81 85 = 5 · 17 131 143 = 11 · 13 181 195 = 3 · 5 · 13

32 33 = 3 · 11 82 91 = 7 · 13 132 133 = 7 · 19 182 183 = 3 · 61

33 85 = 5 · 17 83 105 = 3 · 5 · 7 133 145 = 5 · 29 183 221 = 13 · 17

34 35 = 5 · 7 84 85 = 5 · 17 134 135 = 3³ · 5 184 185 = 5 · 37

35 51 = 3 · 17 85 129 = 3 · 43 135 221 = 13 · 17 185 217 = 7 · 31

36 91 = 7 · 13 86 87 = 3 · 29 136 265 = 5 · 53 186 187 = 11 · 17

37 45 = 3² · 5 87 91 = 7 · 13 137 148 = 2² · 37 187 217 = 7 · 31

38 39 = 3 · 13 88 91 = 7 · 13 138 259 = 7 · 37 188 189 = 3³ · 7

39 95 = 5 · 19 89 99 = 3² · 11 139 161 = 7 · 23 189 235 = 5 · 47

40 91 = 7 · 13 90 91 = 7 · 13 140 141 = 3 · 47 190 231 = 3 · 7 · 11

41 105 = 3 · 5 · 7 91 115 = 5 · 23 141 355 = 5 · 71 191 217 = 7 · 31

42 205 = 5 · 41 92 93 = 3 · 31 142 143 = 11 · 13 192 217 = 7 · 31

43 77 = 7 · 11 93 301 = 7 · 43 143 213 = 3 · 71 193 276 = 2² · 3 · 23

44 45 = 3² · 5 94 95 = 5 · 19 144 145 = 5 · 29 194 195 = 3 · 5 · 13

45 76 = 2² · 19 95 141 = 3 · 47 145 153 = 3² · 17 195 259 = 7 · 37

46 133 = 7 · 19 96 133 = 7 · 19 146 147 = 3 · 7² 196 205 = 5 · 41

47 65 = 5 · 13 97 105 = 3 · 5 · 7 147 169 = 13² 197 231 = 3 · 7 · 11

48 49 = 7² 98 99 = 3² · 11 148 231 = 3 · 7 · 11 198 247 = 13 · 19

49 66 = 2 · 3 · 11 99 145 = 5 · 29 149 175 = 5² · 7 199 225 = 3² · 5²

50 51 = 3 · 17 100 153 = 3² · 17 150 169 = 13² 200 201 = 3 · 67

First few Fermat pseudoprimes in base a (up to 10000)


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a First few Fermat pseudoprimes in base a (up to 10000)OEISsequence

14, 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 composite numbers)

A002808

2 341, 561, 645, 1105, 1387, 1729, 1905, 2047, 2465, 2701, 2821, 3277, 4033, 4369, 4371, 4681, 5461, 6601, 7957, 8321, 8481, 8911 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 A005935

415, 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

A020136

5 4, 124, 217, 561, 781, 1541, 1729, 1891, 2821, 4123, 5461, 5611, 5662, 5731, 6601, 7449, 7813, 8029, 8911, 9881 A005936

635, 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

A005937

7 6, 25, 325, 561, 703, 817, 1105, 1825, 2101, 2353, 2465, 3277, 4525, 4825, 6697, 8321 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

A020137

94, 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

A020138

109, 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 A005939

1110, 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

A020139

1265, 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

A020140

134, 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

A020141

1415, 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

A020142

https://oeis.org/A020142https://oeis.org/A020141https://oeis.org/A020140https://oeis.org/A020139https://oeis.org/A005939https://oeis.org/A020138https://oeis.org/A020137https://oeis.org/A005938https://oeis.org/A005937https://oeis.org/A005936https://oeis.org/A020136https://oeis.org/A005935https://oeis.org/A001567https://oeis.org/A002808


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15 14, 341, 742, 946, 1477, 1541, 1687, 1729, 1891, 1921, 2821, 3133, 3277, 4187, 6541, 6601, 7471, 8701, 8911, 9073 A020143

1615, 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

A020144

174, 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

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 A020146

196, 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

A020147

2021, 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 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 A020149

2221, 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

A020150

2322, 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

A020151

2425, 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

A020152

254, 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

A020153

269, 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 A020154

2726, 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

A020155

289, 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

A020156

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,

https://oeis.org/A020156https://oeis.org/A020155https://oeis.org/A020154https://oeis.org/A020153https://oeis.org/A020152https://oeis.org/A020151https://oeis.org/A020150https://oeis.org/A020149https://oeis.org/A020148https://oeis.org/A020147https://oeis.org/A020146https://oeis.org/A020145https://oeis.org/A020144https://oeis.org/A020143


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29 3523, 4069, 4371, 4411, 5149, 5185, 5356, 5473, 5565, 5611, 6097, 6601, 7161, 7294, 8321, 8401, 8421, 8841, 8911 A020157

3049, 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

A020158

For more information (base 31 to 100), see A020159 to A020228, and for all bases up to 150, see table of Fermat pseudoprimes (text in German)

(http://de.m.wikibooks.org/wiki/Pseudoprimzahlen:_Tabelle_Fermatsche_Pseudoprimzahlen), this page does not define n is a pseudoprime to a base

congruent to 1 or -1 (mod n)

Which bases b make n a Fermat pseudoprime?

The following is a table about all base b < n which n is a Fermat pseudoprime (all composite number is a pseudoprime to base 1, and for b > n, the

solutions are just shifted by k *n for k > 0), if a composite number n is not listed in the table (or n is in the sequence A209211), then n is a pseudoprime

only in base 1, or the bases which congruent to 1 (mod n) (that is, the number of the values of b is 1), these ns are up to 200)

https://oeis.org/A209211http://de.m.wikibooks.org/wiki/Pseudoprimzahlen:_Tabelle_Fermatsche_Pseudoprimzahlenhttps://oeis.org/A020228https://en.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequenceshttps://oeis.org/A020159https://en.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequenceshttps://oeis.org/A020158https://oeis.org/A020157


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n bases b which n is a Fermat pseudoprime (< n)number of the bases of b (< n)(sequence A063994 in OEIS)

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, 50 452 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

https://en.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequenceshttps://oeis.org/A063994


10/14

87 1, 28, 59, 86 4

911, 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 4124 1, 5, 25 3

125 1, 57, 68, 124 4

129 1, 44, 85, 128 4

130 1, 61, 81 3

1331, 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


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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,[7]

this page does not define n is a pseudoprime to a base congruent to 1 or -1 (mod n). Note that when p is aprime, 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, ...(sequence A063994 in OEIS)

https://en.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequenceshttps://oeis.org/A063994https://en.wikipedia.org/wiki/Fermat_little_theoremhttps://en.wikipedia.org/wiki/Wieferich_primehttps://en.wikipedia.org/wiki/If_and_only_if


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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, ... (sequence A105222 in OEIS)

The number of the values of b for n must divides (n), or 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, ... A002997), the quotient = 2 if and only if n

is in the sequence: 4, 6, 15, 91, 703, 1891, 2701, 11305, 12403, 13981, 18721, ... A191311)

The least number with n 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, ... (sequenceA064234 in OEIS) (if and only if n is a nontotient, then the nth term of this sequence is 0)

Weak pseudoprimes

A composite number n which satisfy that bn

= b (mod n) is called weak pseudoprime to baseb, the least weak pseudoprime to base b 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, ...(sequence A000790 in OEIS)

Note that all terms are less than or equal to the smallest Carmichael number, 561. Except for 561, only semiprimes can occur in the above sequence, but

not all semiprimes less than 561 occur, a semiprime pq ( p ≤ q) less than 561 occurs in the above sequences if and only if p - 1 divides q - 1. (see

A108574)

If we require that n > b, they 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, ... (sequence A239293 in OEIS)

Carmichael numbers are weak pseudoprimes to all bases.

Euler–Jacobi pseudoprimes

https://en.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequenceshttps://oeis.org/A239293https://oeis.org/A108574https://en.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequenceshttps://en.wikipedia.org/wiki/If_and_only_ifhttps://en.wikipedia.org/wiki/Semiprimehttps://en.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequenceshttps://oeis.org/A000790https://en.wikipedia.org/wiki/Nontotienthttps://en.wikipedia.org/wiki/If_and_only_ifhttps://en.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequenceshttps://oeis.org/A064234https://oeis.org/A191311https://oeis.org/A002997https://en.wikipedia.org/wiki/Carmichael_numberhttps://en.wikipedia.org/wiki/Primehttps://en.wikipedia.org/wiki/If_and_only_ifhttps://en.wikipedia.org/wiki/Quotienthttps://oeis.org/A000010https://en.wikipedia.org/wiki/Euler_phi_functionhttps://en.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequenceshttps://oeis.org/A105222


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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

1. Desmedt, Yvo (2010). "Encryption Schemes". In Atallah, Mikhail J. & Blanton, Marina. Algorithms and theory of computation handbook: Special topics and

techniques. CRC Press. pp. 10–23. ISBN 978-1-58488-820-8.2. Weisstein, Eric W., "Fermat Pseudoprime" (http://mathworld.wolfram.com/FermatPseudoprime.html), MathWorld .

3. Richard Crandall, Carl Pomerance (2001). "Theorem 3.4.2". Prime Numbers – A Computational Perspective. Springer-Verlag. p. 132.

4. Carl Pomerance; John L. Selfridge, Samuel S. Wagstaff, Jr (July 1980). "The pseudoprimes to 25·109" (PDF). Mathematics of Computation 35 (151): 1003–1026.

doi:10.1090/S0025-5718-1980-0572872-7. Cite uses deprecated parameter |coauthors= (help)

5. W. R. Alford; Andrew Granville; Carl Pomerance (1994). "There are Infinitely Many Carmichael Numbers" (PDF). Annals of Mathematics 139: 703–722.

doi:10.2307/2118576.

6. Sierpinski, W. (1988-02-15), "Chapter V.7", in Ed. A. Schinzel, Elementary Theory of Numbers, North-Holland Mathematical Library (2 Sub ed.), Amsterdam:

North Holland, p. 232, ISBN 9780444866622

7. table of pseudoprimes (text in German) (http://de.m.wikibooks.org/wiki/Pseudoprimzahlen:_Tabelle_Pseudoprimzahlen_(15_-_4999))

External links

W. F. Galway and Jan Feitsma, Tables of pseudoprimes and related data (http://www.cecm.sfu.ca/Pseudoprimes/) (comprehensive list of all

pseudoprimes below 264, including factorization, strong pseudoprimes, and Carmichael numbers)A research for pseudoprime (http://www.numericana.com/answer/pseudo.htm)

Retrieved from "https://en.wikipedia.org/w/index.php?title=Fermat_pseudoprime&oldid=703636929"

https://en.wikipedia.org/w/index.php?title=Fermat_pseudoprime&oldid=703636929http://www.numericana.com/answer/pseudo.htmhttp://www.cecm.sfu.ca/Pseudoprimes/http://de.m.wikibooks.org/wiki/Pseudoprimzahlen:_Tabelle_Pseudoprimzahlen_(15_-_4999)https://en.wikipedia.org/wiki/Special:BookSources/9780444866622https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://www.amazon.com/Elementary-Theory-Numbers-North-Holland-Mathematical/dp/0444866620https://dx.doi.org/10.2307%2F2118576https://en.wikipedia.org/wiki/Digital_object_identifierhttps://en.wikipedia.org/wiki/Annals_of_Mathematicshttp://www.math.dartmouth.edu/~carlp/PDF/paper95.pdfhttps://en.wikipedia.org/wiki/Carl_Pomerancehttps://en.wikipedia.org/wiki/Andrew_Granvillehttps://en.wikipedia.org/wiki/W._R._(Red)_Alfordhttps://en.wikipedia.org/wiki/Help:CS1_errors#deprecated_paramshttps://dx.doi.org/10.1090%2FS0025-5718-1980-0572872-7https://en.wikipedia.org/wiki/Digital_object_identifierhttp://www.math.dartmouth.edu/~carlp/PDF/paper25.pdfhttps://en.wikipedia.org/w/index.php?title=Samuel_S._Wagstaff,_Jr&action=edit&redlink=1https://en.wikipedia.org/wiki/John_L._Selfridgehttps://en.wikipedia.org/wiki/Carl_Pomerancehttps://en.wikipedia.org/wiki/MathWorldhttp://mathworld.wolfram.com/FermatPseudoprime.htmlhttps://en.wikipedia.org/wiki/Eric_W._Weissteinhttps://en.wikipedia.org/wiki/Special:BookSources/978-1-58488-820-8https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://books.google.com/books?id=SbPpg_4ZRGsC&pg=SA10-PA23https://en.wikipedia.org/wiki/Mikhail_Atallahhttps://en.wikipedia.org/wiki/Fermat_primality_testhttps://en.wikipedia.org/wiki/Deterministic_algorithmhttps://en.wikipedia.org/wiki/Primality_testhttps://en.wikipedia.org/wiki/RSA_(algorithm)https://en.wikipedia.org/wiki/Public-key_cryptographyhttps://en.wikipedia.org/wiki/Industrial-grade_primeshttps://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_testhttps://en.wikipedia.org/wiki/Baillie-PSW_primality_testhttps://en.wikipedia.org/wiki/Solovay%E2%80%93Strassen_primality_testhttps://en.wikipedia.org/wiki/Randomized_algorithmhttps://en.wikipedia.org/wiki/Carmichael_numberhttps://en.wikipedia.org/wiki/Euler%E2%80%93Jacobi_pseudoprimehttps://en.wikipedia.org/wiki/Strong_pseudoprime


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Categories: Pseudoprimes Asymmetric-key algorithms

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