Carmichael number
In number theory, a Carmichael number is a composite positive integer n which satisfies the congruence b^{n − 1} ≡ 1 (mod n) for all integers b which are relatively prime to n (see modular arithmetic). They are named for Robert Carmichael.
Contents
Overview
Fermat's little theorem states that all prime numbers have that property. In this sense, Carmichael numbers are similar to prime numbers. They are called pseudoprimes. Carmichael numbers are sometimes also called absolute pseudoprimes.
Carmichael numbers are important because they can fool the Fermat primality test, thus they are always fermat liars. Since Carmichael numbers exist, this primality test cannot be relied upon to prove the primality of a number, although it can still be used to prove a number is composite.
Still, as numbers become larger, Carmichael numbers become very rare. For example, there are 585,355 Carmichael numbers between 1 and 10^{17} (approximately one in 170 billion numbers.) The test is still slightly risky compared to other primality tests such as the Solovay-Strassen primality test.
An alternative and equivalent definition of Carmichael numbers is given by Korselt's theorem from 1899.
Theorem (Korselt 1899): A positive composite integer n is a Carmichael number if and only if n is square-free, and for all prime divisors p of n, it is true that p − 1 divides n − 1.
It follows from this theorem that all Carmichael numbers are odd.
Korselt was the first who observed these properties, but he could not find an example. In 1910 Robert Daniel Carmichael found the first and smallest such number, 561, and hence the name.
That 561 is a Carmichael number can be seen with Korselt's theorem. Indeed, 561 = 3 · 11 · 17 is squarefree and 2 | 560, 10 | 560 and 16 | 560. (The notation a | b means: a divides b).
The next few Carmichael numbers are (sequence A002997 in OEIS):
- 1105 = 5 · 13 · 17 (4 | 1104, 12 | 1104, 16 | 1104)
- 1729 = 7 · 13 · 19 (6 | 1728, 12 | 1728, 18 | 1728)
- 2465 = 5 · 17 · 29 (4 | 2464, 16 | 2464, 28 | 2464)
- 2821 = 7 · 13 · 31 (6 | 2820, 12 | 2820, 30 | 2820)
- 6601 = 7 · 23 · 41 (6 | 6600, 22 | 6600, 40 | 6600)
- 8911 = 7 · 19 · 67 (6 | 8910, 18 | 8910, 66 | 8910)
J. Chernick proved a theorem in 1939 which can be used to construct a subset of Carmichael numbers. The number (6k + 1)(12k + 1)(18k + 1) is a Carmichael number if its three factors are all prime.
Paul Erdős heuristically argued there should be infinitely many Carmichael numbers. In 1994 it was shown by William Alford, Andrew Granville and Carl Pomerance that there really exist infinitely many Carmichael numbers.
It has also been shown that for sufficiently large n, there are at least n^{2/7} Carmichael numbers between 1 and n.
Löh and Niebuhr in 1992 found some of these huge Carmichael numbers including one with 1,101,518 factors and over 16 million digits.
Properties
Carmichael numbers have at least three positive prime factors. The first Carmichael numbers with k = 3, 4, 5, … prime factors are (sequence A006931 in OEIS):
k | |
---|---|
3 | 561 = 3 · 11 · 17 |
4 | 41041 = 7 · 11 · 13 · 41 |
5 | 825265 = 5 · 7 · 17 · 19 · 73 |
6 | 321197185 = 5 · 19 · 23 · 29 · 37 · 137 |
7 | 5394826801 = 7 · 13 · 17 · 23 · 31 · 67 · 73 |
8 | 232250619601 = 7 · 11 · 13 · 17 · 31 · 37 · 73 · 163 |
9 | 9746347772161 = 7 · 11 · 13 · 17 · 19 · 31 · 37 · 41 · 641 |
The first Carmichael numbers with 4 prime factors are (sequence A074379 in OEIS):
i | |
---|---|
1 | 41041 = 7 · 11 · 13 · 41 |
2 | 62745 = 3 · 5 · 47 · 89 |
3 | 63973 = 7 · 13 · 19 · 37 |
4 | 75361 = 11 · 13 · 17 · 31 |
5 | 101101 = 7 · 11 · 13 · 101 |
6 | 126217 = 7 · 13 · 19 · 73 |
7 | 172081 = 7 · 13 · 31 · 61 |
8 | 188461 = 7 · 13 · 19 · 109 |
9 | 278545 = 5 · 17 · 29 · 113 |
10 | 340561 = 13 · 17 · 23 · 67 |
Incidentally, the first Carmichael number (561) is expressible as the sum of two first powers in more ways than any smaller number (although admittedly all numbers share this property). The second Carmichael number (1105) can be expressed as the sum of two squares in more ways than any smaller number. The third Carmichael number (1729) is the Hardy-Ramanujan Number: the smallest number that can be expressed as the sum of two cubes in two different ways.
Higher-order Carmichael numbers
Carmichael numbers can be generalized using concepts of abstract algebra.
The above definition states that a composite integer n is Carmichael precisely when the nth-power-raising function p_{n} from the ring Z_{n} of integers modulo n to itself is the identity function. The identity is the only Z_{n}-algebra endomorphism on Z_{n} so we can restate the definition as asking that p_{n} be an algebra endomorphism of Z_{n}. As above, p_{n} satisfies the same property whenever n is prime.
The nth-power-raising function p_{n} is also defined on any Z_{n}-algebra A. A theorem states that n is prime if and only if all such functions p_{n} are algebra endomorphisms.
In-between these two conditions lies the definition of Carmichael number of order m for any positive integer m as any composite number n such that p_{n} is an endomorphism on every Z_{n}-algebra that can be generated as Z_{n}-module by m elements. Carmichael numbers of order 1 are just the ordinary Carmichael numbers.
Properties
Korselt's criterion can be generalized to higher-order Carmichael numbers, see Howe's paper listed below.
A heuristic argument, given in the same paper, appears to suggest that there are infinitely many Carmichael numbers of order m, for any m. However, not a single Carmichael number of order 3 or above is known.
Layman's overview
To see if a number n is a Carmichael number:
- n must not be prime (must have factors)
- For every number b less than n which has no factors in common with n
- (b^{n − 1}) mod n = 1
The following algorithm (in BASIC) performs this test:
INPUT n n1 = n - 1 fail = 0 somefactor = 0 FOR b = 2 TO n1 IF coprime(b, n) THEN bi = 1 FOR i = 1 TO n1 bi = bi * b bi = bi MOD n NEXT i IF bi <> 1 THEN fail = b EXIT FOR END IF ELSE somefactor = 1 END IF NEXT b IF fail > 0 THEN PRINT n; "fails for b="; fail ELSEIF n <= 1 THEN PRINT n; "is 0 or 1" ELSEIF somefactor = 0 THEN PRINT n; "is a prime" ELSE PRINT n; "is a Carmichael number" END IF
This produces results such as:
560 fails for b= 3 561 is a Carmichael number 562 fails for b= 3 563 is a prime 564 fails for b= 5
References
- Chernick, J. (1935). On Fermat's simple theorem. Bull. Amer. Math. Soc. 45, 269–274.
- Ribenboim, Paolo (1996). The New Book of Prime Number Records.
- Howe, Everett W. (2000). Higher-order Carmichael numbers. Mathematics of Computation 69, 1711–1719. (online version)
- Löh, Günter and Niebuhr, Wolfgang (1996). A new algorithm for constructing large Carmichael numbers(pdf)
- Korselt (1899). Probleme chinois. L'intermediaire des mathematiciens, 6, 142–143.
- Carmichael, R. D. (1912) On composite numbers P which satisfy the Fermat congruence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^{P-1}\equiv 1\bmod P} . Am. Math. Month. 19 22–27.
- Erdős, Paul (1956). On pseudoprimes and Carmichael numbers, Publ. Math. Debrecen 4, 201 –206.
- Alford, Granville and Pomerance (1994). There are infinitely many Carmichael numbers, Ann. of Math. 140(3), 703–722.
External links
de:Carmichael-Zahl es:Número de Carmichael fr:Nombre de Carmichaël ko:카마이클 수 sl:Carmichaelovo število zh:卡邁克爾數