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Pseudoprime

In general, an integer which has a certain property shared by all prime numbers, but is itself not prime, is called a pseudoprime for that particular property.

The most important class of pseudoprimes come from the Fermat's little theorem and hence they are called Fermat pseudoprimes. This theorem states that if p is prime and a is coprime to p, then ap-1 - 1 is divisible by p. If a number x is not prime, a is coprime to x and x divides ax-1 - 1, then x is called a pseudoprime to base a. A number x that is a pseudoprime for all values of a that are coprime to x is called a Carmichael number.

The smallest pseudoprime for the base 2 is 341. It is not a prime, since it equals 11 · 31, nevertheless it satisfies Fermats little theorem; 2341=2 (mod 341).

There are applications, such as public key cryptography like RSA that need large prime numbers. The usual algorithm to generate prime numbers is to generate a random odd numbers and test them for primality. However, primality tests are slow. If the user does not require the test to be completely accurate (say, he might tolerate a very small chance that a composite number is declared to be prime), there are fast algorithms based on Fermat's Little Theorem. For example, a simple way to check whether a number x is prime, is to pick some random number a, and check whether x divides a x - a. If the answer is no, then x cannot be prime, if the answer is yes, x can be called a "probable prime" and the test can be repeated with other values for a. There's a separate article about the Fermat primality test.

Improved versions of this test give strong probable primes. It has been shown by Monier and Rabin that for the improved test that for each a, the chance of catching a false prime is at least 3 in 4.

There are infinitely many pseudoprimes (in fact infinitely many Carmichael numbers), but they are rather rare. There are only 3 pseudo-primes to base 2 below 1000, to a million there are only 245. Pseudoprimes to base 2 are called Poulet numbers or sometimes Sarrus numbers or Fermatians (SIDN A001567) (http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=001567). Poulet numbers and Carmichael numbers (in bold) below 10000 are:

n 
1 341 = 11 · 31
2 561 = 3 · 11 · 17
3 645 = 3 · 5 · 43
4 1105 = 5 · 13 · 17
5 1387 = 19 · 73
6 1729 = 7 · 13 · 19
7 1905 = 3 · 5 · 127
8 2047 = 23 · 89
9 2465 = 5 · 17 · 29
10 2701 = 37 · 73
11 2821 = 7 · 13 · 31
12 3277 = 29 · 112
13 4033 = 37 · 109
14 4369 = 17 · 257
15 4371 = 3 · 31 · 47
16 4681 = 31 · 151
17 5461 = 43 · 127
18 6601 = 7 · 23 · 41
19 7957 = 73 · 109
20 8321 = 53 · 157
21 8481 = 3 · 11 · 257
22 8911 = 7 · 19 · 67

A Poulet number all of whose divisors d divide 2d - 2 is called super-Poulet number. There are an infinitely many Poulet numbers which are not super-Poulet Numbers.

The first smallest pseudoprimes for bases a ≤ 200 are given in the following table; the colors mark the number of prime factors.

asmallest p-pasmallest p-pasmallest p-pasmallest p-p
    51 65 = 5 · 13 101 175 = 52 · 7 151 175 = 52 · 7
2 341 = 11 · 13 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 57 65 = 5 · 13 107 133 = 7 · 19 157 186 = 2 · 3 · 31
8 9 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 · 7 112 121 162 481 = 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 68 69 = 3 · 23 118 119 = 7 · 17 168 169
19 45 = 32 · 5 69 85 = 5 · 17 119 177 = 3 · 59 169 231 = 3 · 7 · 11
20 21 = 3 · 7 70 169 120 121 170 171 = 32 · 19
21 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 74 75 = 3 · 52 124 125 = 33 174 175 = 52 · 7
25 28 = 22 · 7 75 91 = 7 · 13 125 133 = 7 · 19 175 319 = 11 · 19
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 80 81 = 34 130 217 = 7 · 31 180 217 = 7 · 31
31 49 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 · 7 147 169 197 231 = 3 · 7 · 11
48 49 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 · 17 150 169 200 201 = 3 · 67

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