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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 ...
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 ...
The first deterministic primality test significantly faster than the naive methods was the cyclotomy test; its runtime can be proven to be O((log n) c log log log n), where n is the number to test for primality and c is a constant independent of n. A number of further improvements were made, but none could be proven to have polynomial running time.
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 .
ACP 125 (G) Plain-Language Radio Check Procedure Words Proword Meaning Conjunction Proword Meaning LOUD: Your signal is very strong. AND or BUT, depending on which prowords are combined CLEAR: The quality of your transmission is excellent. GOOD: Your signal strength is good. READABLE: The quality of your transmission is satisfactory. WEAK
In probability, weak dependence of random variables is a generalization of independence that is weaker than the concept of a martingale [citation needed].A (time) sequence of random variables is weakly dependent if distinct portions of the sequence have a covariance that asymptotically decreases to 0 as the blocks are further separated in time.
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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 that, in the course of our probable prime tests, we come across two bases a and a ′ for which a 2 r d ≡ a ′ 2 r ′ d ≡ − 1 ( mod n ...