Search results
Results from the WOW.Com Content Network
The Lucas–Lehmer test works as follows. Let M p = 2 p − 1 be the Mersenne number to test with p an odd prime. The primality of p can be efficiently checked with a simple algorithm like trial division since p is exponentially smaller than M p. Define a sequence {} for all i ≥ 0 by
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.
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]
It was eventually determined, after three centuries and the availability of new techniques such as the Lucas–Lehmer test, that Mersenne's conjecture contained five errors, namely two entries are composite (those corresponding to the primes n = 67, 257) and three primes are missing (those corresponding to the primes n = 61, 89, 107).
During the era of manual calculation, all previously untested exponents up to and including 257 were tested with the Lucas–Lehmer test and found to be composite. A notable contribution was made by retired Yale physics professor Horace Scudder Uhler, who did the calculations for exponents 157, 167, 193, 199, 227, and 229. [11]
In 2018, GIMPS adopted a Fermat primality test with basis a=3 [6] [7] as an alternative option for primality testing, [8] while keeping the Lucas-Lehmer test as a double-check for Mersenne numbers detected as probable primes by the Fermat test. [9] (While the Lucas-Lehmer test is deterministic and the Fermat test is only probabilistic, the ...
Lucas test may refer to Lucas primality test for primality of general numbers; Lucas–Lehmer primality test for Mersenne primes; Lucas' reagent, ...