Prime numbers and primality tests
- To: mathgroup at smc.vnet.net
- Subject: [mg129449] Prime numbers and primality tests
- From: johnfeth at gmail.com
- Date: Mon, 14 Jan 2013 23:30:04 -0500 (EST)
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A straightforward way to test a prime number candidate is the Miller-Rabin test (sometimes called the Rabin-Miller test). This well known and popular test is commonly executed 50 times on a candidate prime and has a proven probability of missing a non-prime of no more than 0.25 for each execution. Note that passing 50 Miller-Rabin tests (which is a de facto standard), the probability of non-primality is 0.25^50 ~ 7.9*10^-31, I'm satisfied that the number NextPrime gives me is "prime enough". Mathematica uses the Miller-Rabin test, although it is not clear how many iterations are used. As I understand it, Mathematica also the Lucas pseudo prime test on the Miller-Rabin output. It is interesting to note, however, that the Lucas pseudo prime method of primality testing apparently does not have the handy "feature" of the Miller-Rabin test, namely, the provable, and bounded low probability of a wrong answer, from whence an estimate of primality for any number can be made without finding a counter example! I've read that there are have been no counter-examples (viz., no non-primes that pass the the Lucas pseudo prime test) to numbers that pass the Lucas pseudo prime test, but then again, I've never found an oyster with a pearl inside. Is the Miller-Rabin a better test that the Lucas pseudo prime test? http://reference.wolfram.com/mathematica/tutorial/IntegerAndNumberTheoreticalFunctions.html http://reference.wolfram.com/mathematica/tutorial/SomeNotesOnInternalImplementation.html http://mathworld.wolfram.com/LucasPseudoprime.html