Carmichael number


In number theory, a Carmichael number is a composite number which satisfies the modular arithmetic congruence relation:
for all integers which are relatively prime to.
They are named for Robert Carmichael.
The Carmichael numbers are the subset K1 of the Knödel numbers.
Equivalently, a Carmichael number is a composite number for which
for all integers.

Overview

states that if p is a prime number, then for any integer b, the number b − b is an integer multiple of p. Carmichael numbers are composite numbers which have this property. Carmichael numbers are also called Fermat pseudoprimes or absolute Fermat pseudoprimes. A Carmichael number will pass a Fermat primality test to every base b relatively prime to the number, even though it is not actually prime.
This makes tests based on Fermat's Little Theorem less effective than strong probable prime tests such as the Baillie–PSW primality test and the Miller–Rabin primality test.
However, no Carmichael number is either an Euler–Jacobi pseudoprime or a strong pseudoprime to every base relatively prime to it
so, in theory, either an Euler or a strong probable prime test could prove that a Carmichael number is, in fact, composite.
Arnault
gives a 397-digit Carmichael number that is a strong pseudoprime to all prime bases less than 307:
where
is a 131-digit prime. is the smallest prime factor of, so this Carmichael number is also a pseudoprime to all bases less than.
As numbers become larger, Carmichael numbers become increasingly rare. For example, there are 20,138,200 Carmichael numbers between 1 and 1021.

Korselt's criterion

An alternative and equivalent definition of Carmichael numbers is given by Korselt's criterion.
It follows from this theorem that all Carmichael numbers are odd, since any even composite number that is square-free will have at least one odd prime factor, and thus results in an even dividing an odd, a contradiction.
From the criterion it also follows that Carmichael numbers are cyclic. Additionally, it follows that there are no Carmichael numbers with exactly two prime divisors.

Discovery

Korselt was the first who observed the basic properties of Carmichael numbers, but he did not give any examples. In 1910, Carmichael found the first and smallest such number, 561, which explains the name "Carmichael number".
listed the first seven Carmichael numbers
That 561 is a Carmichael number can be seen with Korselt's criterion. Indeed, is square-free and, and.
The next six Carmichael numbers are :
These first seven Carmichael numbers, from 561 to 8911, were all found by the Czech mathematician Václav Šimerka in 1885. His work, however, remained unnoticed.
J. Chernick proved a theorem in 1939 which can be used to construct a subset of Carmichael numbers. The number 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 W. R. Alford, Andrew Granville and Carl Pomerance used a bound on Olson's constant to show that there really do exist infinitely many Carmichael numbers. Specifically, they showed that for sufficiently large, there are at least Carmichael numbers between 1 and.
Thomas Wright proved that if and are relatively prime,
then there are infinitely many Carmichael numbers in the arithmetic progression,
where.
Löh and Niebuhr in 1992 found some very large Carmichael numbers, including one with 1,101,518 factors and over 16 million digits.
This has been improved to 10,333,229,505 prime factors and 295,486,761,787 digits, so the largest known Carmichael number is much greater than the largest known prime.

Properties

Factorizations

Carmichael numbers have at least three positive prime factors. For some fixed R, there are infinitely many Carmichael numbers with exactly R factors; in fact, there are infinitely many such R.
The first Carmichael numbers with prime factors are :
k
3
4
5
6
7
8
9

The first Carmichael numbers with 4 prime factors are :
i
1
2
3
4
5
6
7
8
9
10

The second Carmichael number can be expressed as the sum of two squares in more ways than any smaller number. The third Carmichael number is the Hardy-Ramanujan Number: the smallest number that can be expressed as the sum of two cubes in two different ways.

Distribution

Let denote the number of Carmichael numbers less than or equal to. The distribution of Carmichael numbers by powers of 10 :

123456789101112131415161718192021
00171643105255646154736058241192794470610521224668358535514016443381806822077720138200


In 1953, Knödel proved the upper bound:
for some constant.
In 1956, Erdős improved the bound to
for some constant. He further gave a heuristic argument suggesting that this upper bound should be close to the true growth rate of.
In the other direction, Alford, Granville and Pomerance proved in 1994 that for sufficiently large X,
In 2005, this bound was further improved by Harman to
who subsequently improved the exponent to.
Regarding the asymptotic distribution of Carmichael numbers, there have been several conjectures. In 1956, Erdős conjectured that there were Carmichael numbers for X sufficiently large. In 1981, Pomerance sharpened Erdős' heuristic arguments to conjecture that there are at least
Carmichael numbers up to, where.
However, inside current computational ranges, these conjectures are not yet borne out by the data.

Generalizations

The notion of Carmichael number generalizes to a Carmichael ideal in any number field K. For any nonzero prime ideal in, we have for all in, where is the norm of the ideal. Call a nonzero ideal in Carmichael if it is not a prime ideal and for all, where is the norm of the ideal. When K is, the ideal is principal, and if we let a be its positive generator then the ideal is Carmichael exactly when a is a Carmichael number in the usual sense.
When K is larger than the rationals it is easy to write down Carmichael ideals in : for any prime number p that splits completely in K, the principal ideal is a Carmichael ideal. Since infinitely many prime numbers split completely in any number field, there are infinitely many Carmichael ideals in. For example, if p is any prime number that is 1 mod 4, the ideal in the Gaussian integers Z is a Carmichael ideal.
Both prime and Carmichael numbers satisfy the following equality:

Lucas–Carmichael number

A positive composite integer is a Lucas–Carmichael number if and only if is square-free, and for all prime divisors of, it is true that. The first Lucas–Carmichael numbers are:

Quasi–Carmichael number

Quasi–Carmichael numbers are squarefree composite numbers n with the property that for every prime factor p of n, p + b divides n + b positively with b being any integer besides 0. If b = −1, these are Carmichael numbers, and if b = 1, these are Lucas–Carmichael numbers. The first Quasi–Carmichael numbers are:

Knödel number

An n-Knödel number for a given positive integer n is a composite number m with the property that each i < m coprime to m satisfies. The n = 1 case are Carmichael numbers.

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 pn from the ring Zn of integers modulo n to itself is the identity function. The identity is the only Zn-algebra endomorphism on Zn so we can restate the definition as asking that pn be an algebra endomorphism of Zn.
As above, pn satisfies the same property whenever n is prime.
The nth-power-raising function pn is also defined on any Zn-algebra A. A theorem states that n is prime if and only if all such functions pn 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 pn is an endomorphism on every Zn-algebra that can be generated as Zn-module by m elements. Carmichael numbers of order 1 are just the ordinary Carmichael numbers.

An order 2 Carmichael number

According to Howe, 17 · 31 · 41 · 43 · 89 · 97 · 167 · 331 is an order 2 Carmichael number. This product is equal to 443,372,888,629,441.

Properties

Korselt's criterion can be generalized to higher-order Carmichael numbers, as shown by Howe.
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.