Carmichael Number - Higher-order Carmichael Numbers

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.

Read more about this topic:  Carmichael Number

Famous quotes containing the words carmichael and/or numbers:

    Each coming together of man and wife, even if they have been mated for many years, should be a fresh adventure; each winning should necessitate a fresh wooing.
    —Marie Carmichael Stopes (1880–1958)

    Out of the darkness where Philomela sat,
    Her fairy numbers issued. What then ailed me?
    My ears are called capacious but they failed me,
    Her classics registered a little flat!
    I rose, and venomously spat.
    John Crowe Ransom (1888–1974)