The Carmichael numbers are the Knödel numbers K < sub > 1 </ sub >.

In this sense, Carmichael numbers are similar to prime numbers ; in fact, they are called Fermat pseudoprimes.

Carmichael numbers are sometimes also called absolute Fermat pseudoprimes.

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

Since Carmichael numbers exist, this primality test cannot be relied upon to prove the primality of a number, although it can still be used to prove a number is composite.

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

For example, there are 20, 138, 200 Carmichael numbers between 1 and 10 < sup > 21 </ sup > ( approximately one in 50 billion numbers ).< ref name =" Pinch2007 "> Richard Pinch, " The Carmichael numbers up to 10 < sup > 21 </ sup >", May 2007 .</ ref > This makes tests based on Fermat's Little Theorem slightly risky compared to others such as the Solovay-Strassen primality test.

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

It follows from this theorem that all Carmichael numbers are odd, since any even composite number that is square-free ( and hence has only one prime factor of two ) will have at least one odd prime factor, and thus results in an even dividing an odd, a contradiction.

( The oddness of Carmichael numbers also follows from the fact that is a Fermat witness for any even number.

From the criterion it also follows that Carmichael numbers are cyclic .< ref > Proof sketch: If is square-free but not cyclic, for two prime factors and of.

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

The next six Carmichael numbers are:

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 ).

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

Whether this formula produces an infinite quantity of Carmichael numbers is an open question ( though it is implied by Dickson's conjecture ).

Ian Carmichael starred as Wimsey in radio adaptations of the novels made by the BBC, all of which have been available on cassette and CD from the BBC Radio Collection.

His Republican opponent, Gil Carmichael, an automobile dealer from Meridian, might have been aided by President Richard Nixon's landslide reelection in forty-nine states, including 78 % of Mississippi's popular vote.

Under Beiderbecke ’ s spell, Carmichael started to play the cornet as well, but found that he didn't have the lips for it, and only played it for a short while.

In 1964, while The Beatles were exploding on the scene, Carmichael lamented, " I'll betcha I have twenty-five songs lying in my trunk " and no one was calling to say " have you got a real good song for such-and such an artist ".

Poirot tells that Carmichael would have been alive, had the Churston letter not gone astray.

Franklin knew that after Lady Clarke's death, Thora and Carmichael would have come closer and ended up in a marriage, as it usually happens with wealthy widowers.

Several people have had the surname Carmichael:

After some unexplained disappearances, Christopher does some exploring and discovers that Mrs. Bouvier and the Reverend Carmichael have some very unwholesome intentions for the young men of their school.

On 9 August 1952 a propeller-driven Sea Fury, piloted by Lieutenant Peter Carmichael of No. 802 Squadron, based on HMS Ocean, shot down a MiG-15 jet fighter, becoming one of only a handful of pilots of propeller planes to have shot down a jet.

Faculty have included Dorothy Allison, Claire Carmichael, Bernard Cooper, Elana Dykewomon, Katherine V. Forrest, Rigoberto Gonzalez, Eloise Klein Healy, Fenton Johnson, Michael Nava, D. A.

Carmichael was eventually released from Turner at the end of 2011, and hopes to have a ride for 2013 Nascar season.

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

Lord Peter Wimsey was played by Ian Carmichael in a series of independent serials that ran from 1972 to 1975 and adapted five novels ( Clouds of Witness, The Unpleasantness at the Bellona Club, Five Red Herrings, Murder Must Advertise and The Nine Tailors ) and by Edward Petherbridge in 1987, in which three of the four major Wimsey / Vane novels ( Strong Poison, Have his Carcase and Gaudy Night ) were dramatised.

" The black contingent at the conference included the American Stokely Carmichael and the British Michael X.

Also notable was the fact that Hoagy Carmichael recorded " Stardust " for the first time in Richmond at the Gennett recording studio.

Carmichael is located at ( 38. 639431 ,-121. 321348 ).

Hoagy Carmichael was the pianist and vocalist at the hotel for two seasons and local legend has it that his hit Stardust was written while at the Keuka hotel.

Richard won, but chairman Leonard Carmichael told him he'd better find something or never " come begging at our door again.

The Unpleasantness at the Bellona Club was adapted for television in 1972, as part of a series starring Ian Carmichael as Lord Peter.

Ian Carmichael died, at the age of 89, at his home in the Esk Valley on the North York Moors on 5 February 2010.

He was named Hoagland after a circus troupe " The Hoaglands " who stayed at the Carmichael house during his mother's pregnancy.

Carmichael earned his first money ($ 5. 00 ) as a musician playing at a fraternity dance that year and began his musical career.

Carmichael finished and recorded one of his most famous songs, " Star Dust " ( later renamed " Stardust ", with Mitchell Parish's lyrics added in 1929 ), at the Gennett Records studio in Richmond, Indiana, with Carmichael doing the piano solo.

Despite his growing fame, at this stage Carmichael was still held back by his inability to sight-read and notate music properly, though innovative for the time.

Fortunately, Louis Armstrong then recorded " Rockin ' Chair " at Okeh studios, giving a badly needed boost to Carmichael.

Carmichael had begun to work at an investment house and was considering a switch in career when he composed " Georgia on My Mind " ( lyrics by Stuart Gorrell ), perhaps most famous in the Ray Charles rendition recorded many years later.

Carmichael started to emerge as a solo singer-performer, first at parties, then professionally.

Hoagy Carmichael at piano, with Lauren Bacall in To Have and Have Not ( film ) | To Have and Have Not

Carmichael appeared as an actor in a total of 14 motion pictures, always performing at least one of his songs, including Young Man with a Horn ( based on friend Bix Beiderbecke's life ) with Bacall and Kirk Douglas, and multi-Academy Award winner The Best Years of Our Lives with Myrna Loy and Fredric March ), in which he teaches a disabled veteran with metal prostheses to play " Chop Sticks ".

Carmichael was a Republican supporter and anti-FDR, voting for Wendell Wilkie for president in 1940, and was often aghast at the left-leaning political views of his friends in Hollywood.

With the Johnny Appleseed Suite, Carmichael once again tried his hand at a longer musical composition, but the episodic treatment lacked the compositional unity and momentum of works such as George Gershwin's Rhapsody in Blue.