Carmichael number

In number theory, a Carmichael number is a composite number ${\displaystyle n}$ which satisfies the modular arithmetic congruence relation:

for all integers ${\displaystyle 1<b<n}$ which are relatively prime to ${\displaystyle n}$. They are named for Robert Carmichael. The Carmichael numbers are the subset K₁ of the Knödel numbers.

Fermat's little theorem states that if p is a prime number, then for any integer b, the number b ^p − b is an integer multiple of p. Carmichael numbers are composite numbers which have the same property of modular arithmetic congruence. In fact, Carmichael numbers are also called Fermat pseudoprimes or 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. This makes tests based on Fermat's Little Theorem risky compared to other more stringent tests such as the Solovay-Strassen primality test or a strong pseudoprime test. Still, as numbers become larger, Carmichael numbers become very rare. For example, there are 20,138,200 Carmichael numbers between 1 and 10²¹ (approximately one in 50 trillion (5*10¹³) numbers).

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