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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.
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.
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 most efficient method presently known for testing the primality of Mersenne numbers is the Lucas–Lehmer primality test. Specifically, it can be shown that for prime p > 2, M p = 2 p − 1 is prime if and only if M p divides S p − 2, where S 0 = 4 and S k = (S k − 1) 2 − 2 for k > 0.
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.
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Lucas sequences are used in some primality proof methods, including the Lucas–Lehmer–Riesel test, and the N+1 and hybrid N−1/N+1 methods such as those in Brillhart-Lehmer-Selfridge 1975. [4] LUC is a public-key cryptosystem based on Lucas sequences [5] that implements the analogs of ElGamal (LUCELG), Diffie–Hellman (LUCDIF), and RSA (LUCRSA
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