Carmichael number

In number theory, a Carmichael number is a composite positive integer $n$ which satisfies the congruence $b^\left\{n-1\right\}~equiv 1 pmod\left\{n\right\}$ for all integers $b$ which are relatively prime to $n$ (see modular arithmetic). They are named for Robert Carmichael. The Carmichael numbers are the Knödel numbers "K"1.

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 Fermat pseudoprimes. Carmichael numbers are sometimes also called absolute Fermat 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 1,401,644 Carmichael numbers between 1 and 1018 (approximately one in 700 billion numbers.) [Richard Pinch, [http://arxiv.org/abs/math/0604376 "The Carmichael numbers up to 1018"] , April 2006 (building on his earlier work [http://www.chalcedon.demon.co.uk/rgep/p37.ps] [http://arxiv.org/abs/math.NT/9803082] [http://arxiv.org/abs/math.NT/0504119] ).] This makes tests based on Fermat's Little Theorem slightly risky compared to others such as the Solovay-Strassen primality test.

An alternative and equivalent definition of Carmichael numbers is given by Korselt's criterion.

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 $p - 1 | n - 1$ (the notation $a | b$ indicates that $a$ divides $b$).

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 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 criterion. Indeed, $561 = 3 cdot 11 cdot 17$ is squarefree and $2 | 560$, $10 | 560$ and $16 | 560$.

The next few Carmichael numbers are OEIS|id=A002997::$1105 = 5 cdot 13 cdot 17 qquad \left(4 mid 1104; 12 mid 1104; 16 mid 1104\right)$:$1729 = 7 cdot 13 cdot 19 qquad \left(6 mid 1728; 12 mid 1728; 18 mid 1728\right)$:$2465 = 5 cdot 17 cdot 29 qquad \left(4 mid 2464; 16 mid 2464; 28 mid 2464\right)$:$2821 = 7 cdot 13 cdot 31 qquad \left(6 mid 2820; 12 mid 2820; 30 mid 2820\right)$:$6601 = 7 cdot 23 cdot 41 qquad \left(6 mid 6600; 22 mid 6600; 40 mid 6600\right)$:$8911 = 7 cdot 19 cdot 67 qquad \left(6 mid 8910; 18 mid 8910; 66 mid 8910\right)$

J. Chernick proved a theorem in 1939 which can be used to construct a subset of Carmichael numbers. The number $\left(6k + 1\right)\left(12k + 1\right)\left(18k + 1\right)$ is a Carmichael number if its three factors are all prime. Whether this formula produces an infinite quantity of Carmichael numbers is an open question.

Paul Erdős heuristically argued there should be infinitely many Carmichael numbers. In 1994 it was shown by W. R. (Red) Alford, Andrew Granville and Carl Pomerance that there really do exist infinitely many Carmichael numbers. Specifically, they showed that for sufficiently large $n$, there are at least $n^\left\{2/7\right\}$ Carmichael numbers between 1 and $n$. [W. R. Alford, A. Granville, and C. Pomerance. [http://www.math.dartmouth.edu/~carlp/PDF/paper95.pdf "There are Infinitely Many Carmichael Numbers."] "Annals of Mathematics" 139 (1994) 703-722.]

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 $k = 3, 4, 5, ldots$ prime factors are OEIS|id=A006931:

As of December 2007, it has been shown that there are 8220777 Carmichael numbers up to 1020.

In the other direction, Alford, Granville and Pomerance proved in their 1994 paper that

:$C\left(X\right) > X^\left\{2/7\right\}$

for sufficiently large $X$ and Glyn Harman proved that

:$C\left(X\right) > X^\left\{0.332\right\},$

again for sufficiently large $X$. [Glyn Harman. "On the number of Carmichael numbers up to X." "Bull. Lond. Math. Soc." 37 (2005) 641-650.] This author has subsequentlyimproved the exponent to just over $1/3$. Erds also gave a heuristic suggesting that his upper bound should be close to the true rate of growth of $C\left(X\right)$.

The distribution of Carmichael numbers by powers of 10, from Pinch (2006).

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 "n"th-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 "n"th-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, as shown by Howe. [Everett W. Howe. [http://arxiv.org/abs/math.NT/9812089 "Higher-order Carmichael numbers."] "Mathematics of Computation" 69 (2000), pp. 1711&ndash;1719.]

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.

References

* Chernick, J. (1935). On Fermat's simple theorem. "Bull. Amer. Math. Soc." 45, 269&ndash;274.
* Ribenboim, Paolo (1996). "The New Book of Prime Number Records".
* Löh, Günter and Niebuhr, Wolfgang (1996). [http://www.ams.org/mcom/1996-65-214/S0025-5718-96-00692-8/S0025-5718-96-00692-8.pdf "A new algorithm for constructing large Carmichael numbers"] (pdf)
* Korselt (1899). Problème chinois. "L'intermédiaire des mathématiciens", 6, 142&ndash;143.
* Carmichael, R. D. (1912) On composite numbers P which satisfy the Fermat congruence . "Am. Math. Month." 19 22&ndash;27.
* Erdős, Paul (1956). On pseudoprimes and Carmichael numbers, "Publ. Math. Debrecen" 4, 201 &ndash;206.

* [http://de.wikibooks.org/wiki/Pseudoprimzahlen:_Tabelle_Carmichael-Zahlen Table of Carmichael numbers]
* [http://www.mathpages.com/home/kmath028.htm Mathpages: The Dullness of 1729]
*
* Richard G.E. Pinch. The Carmichael numbers up to 10 to the 20. [http://www.chalcedon.demon.co.uk/rgep/rcam.html (list of publications)]

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