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
Famous quotes containing the word discovery:
“We are all humiliated by the sudden discovery of a fact which has existed very comfortably and perhaps been staring at us in private while we have been making up our world entirely without it.”
—George Eliot [Mary Ann (or Marian)
“It was one of those evenings when men feel that truth, goodness and beauty are one. In the morning, when they commit their discovery to paper, when others read it written there, it looks wholly ridiculous.”
—Aldous Huxley (1894–1963)
“Your discovery of the contradiction caused me the greatest surprise and, I would almost say, consternation, since it has shaken the basis on which I intended to build my arithmetic.... It is all the more serious since, with the loss of my rule V, not only the foundations of my arithmetic, but also the sole possible foundations of arithmetic seem to vanish.”
—Gottlob Frege (1848–1925)