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For these numbers, repeated application of the Fermat primality test performs the same as a simple random search for factors. While Carmichael numbers are substantially rarer than prime numbers (Erdös' upper bound for the number of Carmichael numbers [ 3 ] is lower than the prime number function n/log(n) ) there are enough of them that Fermat ...
Certain number-theoretic methods exist for testing whether a number is prime, such as the Lucas test and Proth's test. These tests typically require factorization of n + 1, n − 1, or a similar quantity, which means that they are not useful for general-purpose primality testing, but they are often quite powerful when the tested number n is ...
There are 4842 strong pseudoprimes base 2 and 2163 Carmichael numbers below this limit (see Table 1 of [5]). Starting at 17·257, the product of consecutive Fermat numbers is a base-2 pseudoprime, and so are all Fermat composites and Mersenne composites .
Or to put it algebraically, writing the sequence of prime numbers as (p 1, p 2, p 3, ...) = (2, 3, 5, ...), p n is a strong prime if p n > p n − 1 + p n + 1 / 2 . For example, 17 is the seventh prime: the sixth and eighth primes, 13 and 19, add up to 32, and half that is 16; 17 is greater than 16, so 17 is a strong prime. The first few ...
A composite number n is a strong pseudoprime to at most one quarter of all bases below n; [3] [4] thus, there are no "strong Carmichael numbers", numbers that are strong pseudoprimes to all bases. Thus given a random base, the probability that a number is a strong pseudoprime to that base is less than 1/4, forming the basis of the widely used ...
Just as the strong test checks for the existence of more than two square roots of 1 modulo n, two such tests can sometimes check for the existence of more than two square roots of −1. Suppose, in the course of our probable prime tests, come across two bases a and a ′ for which a 2 r d ≡ a ′ 2 r ′ d ≡ − 1 ( mod n ) {\displaystyle a ...
There is also overlap among strong pseudoprimes to different bases. For example, 1373653 is the smallest strong pseudoprime to bases 2 through 4, and 3215031751 is the smallest strong pseudoprime to bases 2 through 10. Arnault [11] gives a 397-digit Carmichael number N that is a strong pseudoprime to all prime bases less than 307.
However the weak law is known to hold in certain conditions where the strong law does not hold and then the convergence is only weak (in probability). See differences between the weak law and the strong law. The strong law applies to independent identically distributed random variables having an expected value (like the weak law).