Carmichael number

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In number theory, a Carmichael number is a composite positive integer n which satisfies the congruence bn − 1 ≡ 1 (mod n) for all integers b which are relatively prime to n (see modular arithmetic). They are named for Robert Carmichael.

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. 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 585,355 Carmichael numbers between 1 and 1017 (approximately one in 170 billion numbers.) The test is still slightly risky compared to other primality tests such as the Solovay-Strassen primality test.

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

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 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 (sequence A002997 in 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)

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 n2/7 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, … 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 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.

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.

Layman's overview

To see if a number n is a Carmichael number:

  • n must not be prime (must have factors)
  • For every number b less than n which has no factors in common with n
    • (bn − 1) mod n = 1

The following algorithm (in BASIC) performs this test:

INPUT n
n1 = n - 1
fail = 0
somefactor = 0
FOR b = 2 TO n1
 IF coprime(b, n) THEN
  bi = 1
  FOR i = 1 TO n1
   bi = bi * b
   bi = bi MOD n
  NEXT i
  IF bi <> 1 THEN
   fail = b
   EXIT FOR
  END IF
 ELSE
  somefactor = 1
 END IF
NEXT b
IF fail > 0 THEN
 PRINT n; "fails for b="; fail
ELSEIF n <= 1 THEN
 PRINT n; "is 0 or 1"
ELSEIF somefactor = 0 THEN
 PRINT n; "is a prime"
ELSE
 PRINT n; "is a Carmichael number"
END IF

This produces results such as:

560 fails for b= 3 
561 is a Carmichael number
562 fails for b= 3 
563 is a prime
564 fails for b= 5

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.
  • Howe, Everett W. (2000). Higher-order Carmichael numbers. Mathematics of Computation 69, 1711–1719. (online version)
  • Löh, Günter and Niebuhr, Wolfgang (1996). A new algorithm for constructing large Carmichael numbers(pdf)
  • Korselt (1899). Probleme chinois. L'intermediaire des mathematiciens, 6, 142–143.
  • Carmichael, R. D. (1912) On composite numbers P which satisfy the Fermat congruence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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.
  • Alford, Granville and Pomerance (1994). There are infinitely many Carmichael numbers, Ann. of Math. 140(3), 703–722.


External links

de:Carmichael-Zahl es:Número de Carmichael fr:Nombre de Carmichaël ko:카마이클 수 sl:Carmichaelovo število zh:卡邁克爾數