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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. Many further improvements were made, but none could be proven to have polynomial running time.
In mathematics, the Lucas–Lehmer test (LLT) is a primality test for Mersenne numbers. The test was originally developed by Édouard Lucas in 1878 [ 1 ] and subsequently proved by Derrick Henry Lehmer in 1930.
As mentioned above, most applications use a Miller–Rabin or Baillie–PSW test for primality. Sometimes a Fermat test (along with some trial division by small primes) is performed first to improve performance. GMP since version 3.0 uses a base-210 Fermat test after trial division and before running Miller–Rabin tests.
In mathematics, the Lucas–Lehmer–Riesel test is a primality test for numbers of the form N = k ⋅ 2 n − 1 with odd k < 2 n. The test was developed by Hans Riesel and it is based on the Lucas–Lehmer primality test. It is the fastest deterministic algorithm known for numbers of that form.
The Miller–Rabin primality test uses the following extension of Fermat's little theorem: [14] If p is an odd prime and p − 1 = 2 s d with s > 0 and d odd > 0, then for every a coprime to p , either a d ≡ 1 (mod p ) or there exists r such that 0 ≤ r < s and a 2 r d ≡ −1 (mod p ) .
This makes the test a fast polynomial-time algorithm. But Fermat numbers grow so rapidly that only a handful of them can be tested in a reasonable amount of time and space. There are some tests for numbers of the form k 2 m + 1, such as factors of Fermat numbers, for primality. Proth's theorem (1878). Let N = k 2 m + 1 with odd k < 2 m.
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 ...
In computational number theory, the Lucas test is a primality test for a natural number n; it requires that the prime factors of n − 1 be already known. [ 1 ] [ 2 ] It is the basis of the Pratt certificate that gives a concise verification that n is prime.