Carmichael numbers

Carmichael numbers have at least three positive prime factors.

From the criterion it also follows that Carmichael numbers are cyclic.

Let denote the number of Carmichael numbers less than or equal to .

Carmichael numbers can be generalized using concepts of abstract algebra.

Still, as numbers become larger, Carmichael numbers become very rare.

Korselt was the first who observed the basic properties of Carmichael numbers, but he could not find any examples.

Regarding the asymptotic distribution of Carmichael numbers, there have been several conjectures.

Carmichael numbers are sometimes also called absolute Fermat pseudoprimes.

Both prime and Carmichael numbers satisfy the following equality:

The special case K are the Carmichael numbers.

So, there exist not so many Carmichael numbers.

Specifically, they showed that for sufficiently large , there are at least Carmichael numbers between 1 and .

Carmichael numbers are important because they pass the Fermat primality test but are not actually prime.

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

The distribution of Carmichael numbers by powers of 10:

We can say that Carmichael numbers are composite numbers that behave a little bit like they would be a prime number.

Paul Erdős heuristically argued there should be infinitely many Carmichael numbers.

These numbers are called Carmichael numbers.

The converse of Fermat's little theorem is not generally true, as it fails for Carmichael numbers.

Carmichael numbers of order 1 are just the ordinary Carmichael numbers.

Carmichael numbers may be strong pseudoprimes to some bases-for example, 561 is a strong pseudoprime to base 50-but not to all bases.

In 1956, Erdős conjectured that there were Carmichael numbers for X sufficiently large.

Korselt's criterion can be generalized to higher-order Carmichael numbers, as shown by Howe.

J. Chernick proved a theorem in 1939 which can be used to construct a subset of Carmichael numbers.

Therefore, there are no (odd) composite n without lots of witnesses, unlike the case of Carmichael numbers for Fermat's test.