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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 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.
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.
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.
New Mersenne primes are found using the Lucas–Lehmer test (LLT), a primality test for Mersenne primes that is efficient for binary computers. [2] The displayed ranks are among indices currently known as of 2022; while unlikely, ranks may change if smaller ones are discovered.
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Hans Ivar Riesel (May 28, 1929 in Stockholm – December 21, 2014) was a Swedish mathematician who discovered the 18th Mersenne prime in 1957 using the computer BESK: [1] [2] 2 3217-1, comprising 969 digits. He held the record for the largest known prime from 1957 to 1961, when Alexander Hurwitz discovered a larger one. [3]
Different subprojects may run on different operating systems, and may have executables for CPUs, GPUs, or both; while running the Lucas–Lehmer–Riesel test, CPUs with Advanced Vector Extensions and Fused Multiply-Add instruction sets will yield the fastest results for non-GPU accelerated workloads.