Number of Carmichaël

In Theory of the numbers, a number of Carmichaël is a whole made up positive N which checks the following property $\ mathcal P$ :

for entire $a$ , one with the congruence $a^n \ equiv has \ pmod n$

(see Arithmetic modular).

Review

According to the Small theorem of Fermat, all the prime numbers check the property

\ mathcal P

. In this direction, the numbers of Carmichaël are similar to the prime numbers and are thus called Pseudopremier S. the numbers of Carmichaël are also called sometimes absolute pseudopremiers .

The numbers of Carmichaël are important because they put in failure the Test of primality of Fermat; thus, they are lying of Fermat . If the numbers of Carmichaël did not exist, this test of primality could always be used to prove that a number is made up.

The larger the numbers become and the rarer the numbers of Carmichaël become, the majority of them return the test of largely useless primality of Fermat compared with the other tests of primality like the Test of primality of Solovay-Strassen. For example, the 646-ème number of Carmichaël is worth 993.905.641 and there exist 105.212 numbers of Carmichaël between 1 and 10¹⁵.

A characterization of the numbers of Carmichaël is given by the theorem of Korselt in 1899.

Theorem (Korselt 1899): A positive entirety made up N is a number of Carmichaël if and only if no Carré of prime number divides N (it is said that N is quadratfrei ) and for each prime factor p of N , the number p − 1 divides N − 1.

It rises from this theorem that all the numbers of Carmichaël are products of at least three odd prime numbers different.

Korselt was the first to observe these properties, but it could not find examples of number of Carmichaël. In 1910, Robert Daniel Carmichaël found smallest of these numbers, 561, and those were named in its honor.

This number of Carmichaël 561 can be checked with the theorem of Korselt. Indeed, 561 = 3 · 11 · 17 is not divisible by a square from prime number, and 3 - 1 = 2,11 - 1 = 10 and 17 - 1 = 16 are all three of the dividers of 560.

The first numbers of Carmichaël are:

561 = 3 · 11 · 17

1.105 = 5 · 13 · 17

1.729 = 7 · 13 · 19

2.465 = 5 · 17 · 29

2.821 = 7 · 13 · 31

6.601 = 7 · 23 · 41

8.911 = 7 · 19 · 67

The action pursuant of the first 33 numbers of Carmichaël in ascending order is taken thereafter n° A002997 of the electronic Encyclopédie of the whole continuations (OEIS), and the action pursuant of the first 19.279 numbers of Carmichaël (classified by order ascending and broken up into their factors first) is taken here.

J. Chernick showed a theorem in 1939 which can be used to build a subset of numbers of Carmichaël. The number (6 K + 1) (12 K + 1) (18 K + 1) is a number of Carmichaël if its three factors are very first.

Paul Erdős supported in a heuristic way which it should exist there an infinity of numbers of Carmichaël. This conjecture was shown in 1994 by William Alford, Andrew Granville and Carl Pomerance, and even, more precisely, for a sufficiently large N , there exists at least N ^2/7 numbers of Carmichaël ranging between 1 and N .

Richard G.E. Pinch showed when with him that for all N , it does not have there more $n* {exp ({- {\ frac {\ ln \ left (N \ right) \ ln \ left (\ ln \ left (\ ln
\ left (N \ right) \ right) \ right)} {\ ln \ left (\ ln \ left (N
\ right) \ right)}}}})$ numbers of Carmichaël ranging between 1 and N .

Properties

The numbers of Carmichaël have at least three factors first.

The first numbers of Carmichaël with respectively at least K = 3,4,5,… factors first are (A006931 continuation of OEIS):

The first numbers of Carmichaël with four factors first are (A074379 continuation of OEIS):

An amusing coincidence is the following one: the third number of Carmichaël, 1729, is not other than the Nombre of Hardy-Ramanujan, i.e. the smallest positive entirety which can be written in two ways different as the sum from two cubes (1729 = 10³ + 9³ = 12³ + 1³). In the same vein, the second number of Carmichaël, 1105, can be written as summons of two squares of more than ways than any entirety which is lower to him. To enclose the sequence, the first number of Carmichaël 561 can (like any natural entirety) be written like summons unary powers of positive entireties of more than ways than any smaller positive entirety.

Demonstration of the theorem of Korselt

That is to say p a prime number which divides a number of Carmichaël N . There is ( p +1) ^N=1+ Np +C_N² p ²+… ≡1 (MOD p ²). In addition ( p +1) ^N≡ p ^N+1 ≡ p + 1 (MOD N ) (the second congruence rises owing to the fact that N is a number of Carmichael). If p ² divided N , there would be 1≡ ( p +1) ^N≡ p +1 (MOD p ²) what is impossible. That shows that the square of p does not divide N .

N being a number of Carmichael, is has first with N one a:

$a^ {n-1} \ equiv 1 \ pmod {N}$ thus $a^ {n-1} \ equiv 1 \ pmod {p}$ because p divides N . In addition, by the small Theorem of Fermat:

$a^ {p-1} \ equiv 1 \ pmod {p}$ .

One from of deduced immediately that N -1 is a multiple of p -1.

Reciprocally, let us suppose that N is a product of prime numbers all distinct, p ₁, p ₂,… p _K and that the numbers p ₁-1, p ₂-1,… divide all N -1. Then for entire has and all p _I one has N ≡1 (MOD p _I-1) and thus has ≡ has ^{p _I} ≡ has ^{2 p _I-1} ≡ has ^{3 p _I-2}≡… ≡ has ^N (MOD p _I). The number has ^N is adequate with has modulo each p _I, therefore also modulo their product N . It is true for entire has , therefore N is a number of Carmichaël.

This completes the demonstration of the theorem of Korselt.

Consequences of the theorem of Korselt:

That is to say N a number of Carmichaël. It is divisible by several prime numbers disctints, therefore one of them at least is different from two. Let us call p this prime factor odd of N . Since p -1 is even, its multiple N -1 is too. That shows that any number of Carmichaël is odd.

If p is a factor first of a number of Carmichaël N , then modulo p -1 one has p ≡1≡ N and thus 1≡ ( N / p ) p ≡ ( N / p ) 1= N / p . In other words, if p is a factor first of a number of Carmichaël, then the product of the other factors first is adequate to 1 modulo p -1.

A number of Carmichaël cannot be the product of two prime numbers p and Q , because then each of the two numbers p -1 and Q -1 would divide the other and they would be equal.

Any number of Carmichaël is thus the product of at least three odd prime numbers distinct.

Higher orders of the numbers of Carmichaël

The numbers of Carmichaël can be generalized by using the concepts of the general Algèbre.

The definition above states that a made up entirety N is a number of Carmichaël precisely when the function N ième power p _N of the ring Z _N of the entireties modulo N in itself is the function identity. The identity is only Z _N- Endomorphisme of algebra on Z _N thus we can restore the definition by requiring that p _N be a endomorphism of algebra of Z _N. As above, p _N satisfy very the properties when N is first.

The function N ième power p _N is also defined on any Z _N-algèbre has . A theorem states that N is first if and only if all the functions such as p _N are endomorphisms of algebras.

Between these two conditions the definition of the number of Carmichaël of order m for any positive entirety is m as any made up number N such as p _N is a endomorphism on each Z _N-algèbre which can be generated like a Z _N- module by m elements. The numbers of Carmichaël of order 1 are simply the ordinary numbers of Carmichaël.

Properties

The criterion of Korselt can be generalized Carmichaël of raised natures, to see the article of Howe below.

A heuristic argument, given in the same article, seems to suggest that there exists an infinity of numbers of Carmichaël of order m , whatever the m . Nevertheless, no number of single Carmichaël of order 3 or above is known.

References

Chernick, J. (1935). One Fermat' S simple theorem. Bull. Land-mark. Maths. Plowshare 45 , 269-274.
Howe, Everett W. (2000). Higher-order Carmichael numbers. Mathematics off Computation 69 , 1711-1719. (online version)

External bond

Wikisource:Carmichael_numbers
Mathpages: The Dullness off 1729

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