Discovery
Korselt was the first who observed the basic properties of Carmichael numbers, but he could not find 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, is square-free and, and .
The next six Carmichael numbers are (sequence A002997 in OEIS):
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 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, there are at least Carmichael numbers between 1 and .
Löh and Niebuhr in 1992 found some huge Carmichael numbers, including one with 1,101,518 factors and over 16 million digits.
Read more about this topic: Carmichael Number
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