Carmichael number

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. If Carmichael numbers did not exist, this primality test could always be used to prove compositeness of a number.

Even though, as numbers become larger, Carmichael numbers become rarer, the sheer number of them makes the Fermat primality test largely useless compared to other primality tests such as the Solovay-Strassen primality test. For example, there are 105,212 Carmichael numbers between 1 and 10¹⁵.

An alternative and equivalent definition of Carmichael numbers is given by Korselt's theorem from 1899.

Theorem (Korselt 1899): A positive and odd 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 :
: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.

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.

References