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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. If a composite integer x is coprime to an integer a > 1 and 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 Fermat primality test for the base a.[1]

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 Fermat primality test for the base 2.

Pseudoprimes to base 2 are sometimes called Poulet numbers, Sarrus numbers, or Fermatians (sequence A001567 in OEIS).

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 \( a^{q-1} \equiv 1 \pmod q \) for a=q-1. 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 \( a^{q-1} \equiv 1 \pmod q \) and \( 2 \le a \le q-2. \)

This stronger definition excludes every power of three (9, 27, 81, 243, ...), and many even integers from the base-2 Fermat pseudoprimes.

Properties
Distribution

There are infinitely many pseudoprimes to a given base (in fact, infinitely many Carmichael numbers), but they are rather rare. There are only three pseudo-primes to base 2 below 1000, and below a million there are only 245. Starting at 17·257, the product of consecutive Fermat numbers is a base-2 pseudo-prime.
Factorizations

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

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

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

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

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 2d − 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.
a smallest p-p a smallest p-p a smallest p-p a smallest p-p

table.

a	smallest p-p	a	smallest p-p	a	smallest p-p	a	smallest p-p
		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 · 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 = 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
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

Euler–Jacobi pseudoprimes
Main article: 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 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

^ a b 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.
^ Weisstein, Eric W., "Fermat Pseudoprime" from MathWorld.
^ Richard Crandall, Carl Pomerance (2001). "Theorem 3.4.2". Prime Numbers – A Computational Perspective. Springer-Verlag. p. 132.

External links

W. F. Galway, Database of Base-2 Pseudoprimes up to 10^15 (including Strong Pseudoprimes and Carmichael numbers)

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