| Carmichael Number |
Article Index for Carmichael |
Website Links For Carmichael |
Information AboutCarmichael Number |
| CATEGORIES ABOUT CARMICHAEL NUMBER | |
| integer sequences | |
| modular arithmetic | |
| pseudoprimes | |
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for all integers ''b'' which are Relatively Prime to ''n'' (see Modular Arithmetic ). They are named for Robert Carmichael . OVERVIEW Fermat's Little Theorem states that all Prime Numbers have that property. In this sense, Carmichael numbers are similar to prime numbers. They are called Pseudoprime s. Carmichael numbers are sometimes also called absolute pseudoprimes. Carmichael numbers are important because they can fool the Fermat Primality Test , thus they are always ''fermat liars''. 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 585,355 Carmichael numbers between 1 and 1017 (approximately one in 170 billion numbers.) The test is still slightly risky compared to other primality tests such as the Solovay-Strassen Primality Test . An alternative and equivalent definition of Carmichael numbers is given by Korselt's theorem from 1899 . Theorem (Korselt 1899): A positive composite integer ''n'' is a Carmichael number if and only if ''n'' is Square-free , and for all prime divisors ''p'' of ''n'', it is true that ''p'' − 1 divides ''n'' − 1. It follows from this theorem that all Carmichael numbers are Odd . Korselt was the first who observed these properties, but he could not find an example. In 1910 Robert Daniel Carmichael found the first and smallest such number, 561 , and hence the name. | ||
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