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Using fast algorithms for modular exponentiation and multiprecision multiplication, the running time of this algorithm is O(k log 2 n log log n) = Õ(k log 2 n), where k is the number of times we test a random a, and n is the value we want to test for primality; see Miller–Rabin primality test for details.
The simplest probabilistic primality test is the Fermat primality test (actually a compositeness test). It works as follows: Given an integer n, choose some integer a coprime to n and calculate a n − 1 modulo n. If the result is different from 1, then n is composite. If it is 1, then n may be prime.
The false statement that all numbers that pass the Fermat primality test for base 2 are prime is called the Chinese hypothesis. The smallest base-2 Fermat pseudoprime is 341. It is not a prime, since it equals 11·31, but it satisfies Fermat's little theorem: 2 340 ≡ 1 (mod 341) and thus passes the Fermat primality test for the base 2.
Probable primality is a basis for efficient primality testing algorithms, which find application in cryptography. These algorithms are usually probabilistic in nature. The idea is that while there are composite probable primes to base a for any fixed a , we may hope there exists some fixed P <1 such that for any given composite n , if we choose ...
Air Force Two is the air traffic control designated call sign held by any United States Air Force aircraft carrying the vice president of the United States, but not the president. [ 1 ] [ 2 ] The term is often associated with the Boeing C-32 , a modified 757 which is most commonly used as the vice president's transport.
The Miller–Rabin primality test or Rabin–Miller primality test is a probabilistic primality test: an algorithm which determines whether a given number is likely to be prime, similar to the Fermat primality test and the Solovay–Strassen primality test. It is of historical significance in the search for a polynomial-time deterministic ...
As is so often the case, an already big number kept getting bigger and by 2019, Air Force Magazine was reporting that the new planes were expected to cost $5.2 billion. Their 100,000-plus page ...
The first step is for efficiency only. The Baillie–PSW test works without this step, but if n has small prime factors, then the quickest way to show that n is composite is to find a factor by trial division. Step 2 is, in effect, a single application of the Miller–Rabin primality test, but using the fixed base 2.