Carmichael number

Overview

Fermat's little theorem states that if p is a prime number, then for any integer b, the number b ^p − b is an integer multiple of p. Carmichael numbers are composite numbers which have the same property of modular arithmetic congruence. In fact, Carmichael numbers are also called Fermat pseudoprimes or absolute Fermat pseudoprimes. Carmichael numbers are important because they pass the Fermat primality test but are not actually prime. 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. This makes tests based on Fermat's Little Theorem risky compared to other more stringent tests such as the Solovay-Strassen primality test or a strong pseudoprime test. Still, as numbers become larger, Carmichael numbers become very rare. For example, there are 20,138,200 Carmichael numbers between 1 and 10²¹ (approximately one in 50 trillion (5*10¹³) numbers).

Korselt's criterion

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

Theorem (A. 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 .

It follows from this theorem that all Carmichael numbers are odd, since any even composite number that is square-free (and hence has only one prime factor of two) will have at least one odd prime factor, and thus p − 1 ∣ n − 1 results in an even dividing an odd, a contradiction. (The oddness of Carmichael numbers also follows from the fact that − 1 is a Fermat witness for any even composite number.) From the criterion it also follows that Carmichael numbers are cyclic. Additionally, it follows that there are no Carmichael numbers with exactly two prime factors.

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".

That 561 is a Carmichael number can be seen with Korselt's criterion. Indeed, 561 = 3 ⋅ 11 ⋅ 17 is square-free and 2 ∣ 560 , 10 ∣ 560 and 16 ∣ 560 .

The next six Carmichael numbers are (sequence A002997 in the 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 ) .

These first seven Carmichael numbers, from 561 to 8911, were all found by the Czech mathematician Václav Šimerka in 1885 (thus preceding not just Carmichael but also Korselt, although Šimerka did not find anything like Korselt's criterion). 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 ( 6 k + 1 ) ( 12 k + 1 ) ( 18 k + 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 (though it is implied by Dickson's conjecture).

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 very large Carmichael numbers, including one with 1,101,518 factors and over 16 million digits.

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 k = 3 , 4 , 5 , … prime factors are (sequence A006931 in the OEIS):

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

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 (of positive numbers) in two different ways.

Distribution

Let C ( X ) denote the number of Carmichael numbers less than or equal to X . The distribution of Carmichael numbers by powers of 10 (sequence A055553 in the OEIS):

In 1953, Knödel proved the upper bound:

C ( X ) < X exp ⁡ ( − k 1 ( log ⁡ X log ⁡ log ⁡ X ) 1 2 )

for some constant k 1 .

In 1956, Erdős improved the bound to

C ( X ) < X exp ⁡ ( − k 2 log ⁡ X log ⁡ log ⁡ log ⁡ X log ⁡ log ⁡ X )

for some constant k 2 . He further gave a heuristic argument suggesting that this upper bound should be close to the true growth rate of C ( X ) . The table below gives approximate minimal values for the constant k in the Erdős bound for X = 10 n as n grows:

In the other direction, Alford, Granville and Pomerance proved in 1994 that for sufficiently large X,

C ( X ) > X 2 7 .

In 2005, this bound was further improved by Harman to

C ( X ) > X 0.332

and then has subsequently improved the exponent to just over 1 / 3 .

Regarding the asymptotic distribution of Carmichael numbers, there have been several conjectures. In 1956, Erdős conjectured that there were X 1 − o ( 1 ) Carmichael numbers for X sufficiently large. In 1981, Pomerance sharpened Erdős' heuristic arguments to conjecture that there are

X 1 − { 1 + o ( 1 ) } log ⁡ log ⁡ log ⁡ X log ⁡ log ⁡ X

Carmichael numbers up to X. However, inside current computational ranges (such as the counts of Carmichael numbers performed by Pinch up to 10²¹), 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 p in O K , we have α N ( p ) ≡ α mod p for all α in O K , where N ( p ) is the norm of the ideal p . (This generalizes Fermat's little theorem, that m p ≡ m mod p for all integers m when p is prime.) Call a nonzero ideal a in O K Carmichael if it is not a prime ideal and α N ( a ) ≡ α mod a for all α ∈ O K , where N ( a ) is the norm of the ideal a . When K is Q , the ideal a is principal, and if we let a be its positive generator then the ideal a = ( a ) 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 O K : for any prime number p that splits completely in K, the principal ideal p O K is a Carmichael ideal. Since infinitely many prime numbers split completely in any number field, there are infinitely many Carmichael ideals in O K . For example, if p is any prime number that is 1 mod 4, the ideal (p) in the Gaussian integers Z[i] is a Carmichael ideal.

Both prime and Carmichael numbers satisfy the following equality:

gcd ( ∑ x = 1 n − 1 x n − 1 , n ) = 1

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.

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.