- Discrete Weighted Transforms and Large Integer Arithmetic, Richard Crandall and Barry Fagin, 1994
- Rapid Multiplication Modulo the Sum And Difference of Highly Composite Numbers, Colin Percival, 2002
- An FFT Extension to the P-1 Factoring Algorithm, Montgomerry & Silverman, 1990
- Improved Stage 2 to P+/-1 Factoring Algorithms, Montgomerry & Kruppa, 2008
(pronounced purrple categorizer)
PRPLL implements two primality tests for Mersenne numbers: PRP ("PRobable Prime") and LL ("Lucas-Lehmer") as the name suggests.
PRPLL is an OpenCL (GPU) program for primality testing Mersenne numbers.
Invoke make in the source directory.
See prpll -h for the command line options.
For Mersenne primes search, the PRP test is by far preferred over LL, such that LL is not used anymore for search. But LL is still used to verify a prime found by PRP (which is a very rare occurence).
This is a test that proves whether a Mersenne number is prime or not, but without providing a factor in the case where it is not prime. The Lucas-Lehmer test is very simple to describe: iterate the function f(x)=(x^2 - 2) modulo M(p) starting with the number 4. If after p-2 iterations the result is 0, then M(p) is certainly prime, otherwise M(p) is certainly not prime.
Lucas-Lehmer, while a very efficient primality test, still takes a rather long time for large Mersenne numbers (on the order of weeks of intense compute), thus it is only applied to the Mersenne candidates that survived the cheaper preliminary filters TF and P-1.
The probable prime test can prove that a candidate is composite (without providing a factor), but does not prove that a candidate is prime (only stating that it probably is prime) -- although in practice the difference between probable prime and proved prime is extremely small for large Mersenne candidates.
The PRP test is very similar computationally to LL: PRP iterates f(x) = x^2 modulo M(p) starting from 3. If after p iterations the result is 9 modulo M(p), then M(p) is probably prime, otherwise M(p) is certainly not prime. The cost of PRP is exactly the same as LL.