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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.
These numbers have been proved prime by computer with a primality test for their form, for example the Lucas–Lehmer primality test for Mersenne numbers. “!” is the factorial, “#” is the primorial, and () is the third cyclotomic polynomial, defined as + +.
A Lucas probable prime test is most useful if D is chosen such that the Jacobi symbol () is −1 (see pages 1401–1409 of, [1] page 1024 of, [5] or pages 266–269 of [2]). This is especially important when combining a Lucas test with a strong pseudoprime test, such as the Baillie–PSW primality test.
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