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The Lucas-Lehmer residue calculated with these alternative starting values will still be zero if M p is a Mersenne prime. However, the terms of the sequence will be different and a non-zero Lucas-Lehmer residue for non-prime M p will have a different numerical value from the non-zero value calculated when s 0 = 4.
In mathematics, a Lehmer sequence (,) or (,) is a generalization of a Lucas sequence (,) or (,), allowing the square root of an integer R in place of the integer P. [1]To ensure that the value is always an integer, every other term of a Lehmer sequence is divided by √ R compared to the corresponding Lucas sequence.
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
In number theory, Carmichael's theorem, named after the American mathematician R. D. Carmichael, states that, for any nondegenerate Lucas sequence of the first kind U n (P, Q) with relatively prime parameters P, Q and positive discriminant, an element U n with n ≠ 1, 2, 6 has at least one prime divisor that does not divide any earlier one except the 12th Fibonacci number F(12) = U 12 (1, − ...
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.
New Mersenne primes are found using the Lucas–Lehmer test (LLT), a primality test for Mersenne primes that is efficient for binary computers. [2] Due to this efficiency, the largest known prime number has often been a Mersenne prime. [12]
The Lucas–Lehmer–Riesel test is a particular case of group-order primality testing; we demonstrate that some number is prime by showing that some group has the order that it would have were that number prime, and we do this by finding an element of that group of precisely the right order.
This may stand forever as the largest prime number proven by hand. Later Derrick Henry Lehmer refined Lucas's primality tests and obtained the Lucas–Lehmer primality test. He worked on the development of the umbral calculus. Lucas is credited as the first to publish the Kempner function. [6] Lucas was also interested in recreational mathematics.