Carmichael number

In number theory, a Carmichael number is a composite positive integer $n$ which satisfies the congruence $b^{n-1}~\equiv 1{\pmod {n}}$ 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 K₁.

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.) 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 (sequence A002997 in the OEIS):

$1105=5\cdot 13\cdot 17\qquad (4|1104,12|1104,16|1104)$

$1729=7\cdot 13\cdot 19\qquad (6|1728,12|1728,18|1728)$

$2465=5\cdot 17\cdot 29\qquad (4|2464,16|2464,28|2464)$

$2821=7\cdot 13\cdot 31\qquad (6|2820,12|2820,30|2820)$

$6601=7\cdot 23\cdot 41\qquad (6|6600,22|6600,40|6600)$

$8911=7\cdot 19\cdot 67\qquad (6|8910,18|8910,66|8910)$

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. 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 $n^{2/7}$ 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 $k=3,4,5,\ldots$ prime factors are (sequence A006931 in the OEIS):

k
3	$561=3\cdot 11\cdot 17$
4	$41041=7\cdot 11\cdot 13\cdot 41$
5	$825265=5\cdot 7\cdot 17\cdot 19\cdot 73$
6	$321197185=5\cdot 19\cdot 23\cdot 29\cdot 37\cdot 137$
7	$5394826801=7\cdot 13\cdot 17\cdot 23\cdot 31\cdot 67\cdot 73$
8	$232250619601=7\cdot 11\cdot 13\cdot 17\cdot 31\cdot 37\cdot 73\cdot 163$
9	$9746347772161=7\cdot 11\cdot 13\cdot 17\cdot 19\cdot 31\cdot 37\cdot 41\cdot 641$

The first Carmichael numbers with 4 prime factors are (sequence A074379 in the OEIS):

i
1	$41041=7\cdot 11\cdot 13\cdot 41$
2	$62745=3\cdot 5\cdot 47\cdot 89$
3	$63973=7\cdot 13\cdot 19\cdot 37$
4	$75361=11\cdot 13\cdot 17\cdot 31$
5	$101101=7\cdot 11\cdot 13\cdot 101$
6	$126217=7\cdot 13\cdot 19\cdot 73$
7	$172081=7\cdot 13\cdot 31\cdot 61$
8	$188461=7\cdot 13\cdot 19\cdot 109$
9	$278545=5\cdot 17\cdot 29\cdot 113$
10	$340561=13\cdot 17\cdot 23\cdot 67$

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.

Distribution

Let $C(X)$ denote the number of Carmichael numbers less than or equal to $X$. Erdős proved in his 1956 paper that

$C(X)<X\exp {\frac {-k\log {X}\log {\log {\log {X}}}}{\log {\log {X}}}}$

for some constant $k$;

The table below gives approximate values for this constant:

n	k
104	2,19547
106	1,97946
108	1,90495
1010	1,86870
1012	1,86377
1014	1,86293
1016	1,86406
1018	1,86522
1020	1,86598

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

$C(X)>X^{2/7}$

for sufficiently large $X$ and Glyn Harman proved that

$C(X)>X^{0.332},$

again for sufficiently large $X$. This author has subsequently improved the exponent to just over $1/3$. Erdős also gave a heuristic suggesting that his upper bound should be close to the true rate of growth of $C(X)$.

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

$n$	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
$C(10^{n})$	1	7	16	43	105	255	646	1547	3605	8241	19279	44706	105212	246683	585355	1401644	3381806	8220777

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, 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.

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.
• Löh, Günter and Niebuhr, Wolfgang (1996). A new algorithm for constructing large Carmichael numbers(pdf)
• Korselt (1899). Problème chinois. L'intermédiaire des mathématiciens, 6, 142–143.
• Carmichael, R. D. (1912) On composite numbers P which satisfy the Fermat congruence $a^{P-1}\equiv 1{\bmod {P}}$. Am. Math. Month. 19 22–27.
• Erdős, Paul (1956). On pseudoprimes and Carmichael numbers, Publ. Math. Debrecen 4, 201 –206.

External links

• Table of Carmichael numbers
• Mathpages: The Dullness of 1729
• Weisstein, Eric W. "Carmichael Number". MathWorld.
• Final Answers Modular Arithmetic
• Richard G.E. Pinch. The Carmichael numbers up to 10 to the 20. (list of publications)