NAME ptest - probabilistic test of primality SYNOPSIS ptest(n [,count [,skip]]) TYPES n integer count integer with absolute value less than 2^24, defaults to 1 skip integer, defaults to 1 return 0 or 1 DESCRIPTION In ptest(n, ...) the sign of n is ignored; here we assume n >= 0. ptest(n, count, skip) always returns 1 if n is prime; equivalently, if 0 is returned then n is not prime. If n is even, 1 is returned only if n = 2. If count >= 0 and n < 2^32, ptest(n,...) essentially calls isprime(n) and returns 1 only if n is prime. If count >= 0, n > 2^32, and n is divisible by a prime <= 101, then ptest(n,...) returns 0. If count is zero, and none of the above cases have resulted in 0 being returned, 1 is returned. In other cases (which includes all cases with count < 0), tests are made for abs(count) bases b: if n - 1 = 2^s * m where m is odd, the test for base b of possible primality is passed if b is a multiple of n or b^m = 1 (mod n) or b^(2^j * m) = n - 1 (mod n) for some j where 0 <= j < s; integers that pass the test are called strong probable primes for the base b; composite integers that pass the test are called strong pseudoprimes for the base b; ( XXX ) Since the test for base b depends on b % n, and bases 0, 1 and n - 1 are trivial (n is always a strong probable prime for these bases), it is sufficient to consider 1 < b < n - 1. The bases for ptest(n, count, skip) are selected as follows: skip = 0: random in [2, n-2] skip = 1: successive primes 2, 3, 5, ... not exceeding min(n, 65536) otherwise: successive integers skip, skip + 1, ..., skip+abs(count)-1 In particular, if m > 0, ptest(n, -m, 2) == 1 shows that n is either prime or a strong pseudoprime for all positive integer bases <= m + 1. If 1 < b < n - 1, ptest(n, -1, b) == 1 if and only if n is a strong pseudoprime for the base b. For the random case (skip = 0), the probability that any one test with random base b will return 1 if n is composite is always less than 1/4, so with count = k, the probability is less than 1/4^k. For most values of n the probability is much smaller, possible zero. RUNTIME If n is composite, ptest(n, 1, skip) is usually faster than ptest(n, -1, skip), much faster if n is divisible by a small prime. If n is prime, ptest(n, -1, skip) is usually faster than ptest(n, 1, skip), possibly much faster if n < 2^32, only slightly faster if n > 2^32. If n is a large prime (say 50 or more decimal digits), the runtime for ptest(n, count, skip) will usually be roughly K * abs(count) * ln(n)^3 for some constant K. ( XXX ) For composite n with highbit(n) = N, the compositeness is detected quickly if n is divisible by a small prime and count >= 0; otherwise, if count is not zero, usually only one test is required to establish compositeness, so the runtime will probably be about K * N^3. For some rare values of composite n, high values of count may be required to establish the compositeness. If the word-count for n is less than conf("redc2"), REDC algorithms are used in evaluating ptest(n, count, skip) when small-factor cases have been eliminated. For small word-counts (say < 10) this may more than double the speed of evaluation compared with the standard algorithms. EXAMPLE ; print ptest(103^3 * 3931, 0), ptest(4294967291,0) 1 1 In the first example, the first argument > 2^32; in the second the first argument is the largest prime less than 2^32. ; print ptest(121,-1,2), ptest(121,-1,3), ptest(121,-2,2) 0 1 0 121 is the smallest strong pseudoprime to the base 3. ; x = 151 * 751 * 28351 ; print x, ptest(x,-4,1), ptest(x,-5,1) 3215031751 1 0 The integer x in this example is the smallest positive integer that is a strong pseudoprime to each of the first four primes 2, 3, 5, 7, but not to base 11. The probability that ptest(x,-1,0) will return 1 is about .23. ; for (i = 0; i < 11; i++) print ptest(91,-1,0),:; print; 0 0 0 1 0 0 0 0 0 0 1 The results for this example depend on the state of the random number generator; the expectation is that 1 will occur twice. ; a = 24444516448431392447461 * 48889032896862784894921; ; print ptest(a,11,1), ptest(a,12,1), ptest(a,20,2), ptest(a,21,2) 1 0 1 0 These results show that a is a strong pseudoprime for all 11 prime bases less than or equal to 31, and for all positive integer bases less than or equal to 21, but not for the bases 37 and 22. The probability that ptest(a,-1,0) (or ptest(a,1,0)) will return 1 is about 0.19. LIMITS none LINK LIBRARY BOOL qprimetest(NUMBER *n, NUMBER *count, NUMBER *skip) BOOL zprimetest(ZVALUE n, long count, long skip) SEE ALSO factor, isprime, lfactor, nextcand, nextprime, prevcand, prevprime, pfact, pix ## Copyright (C) 1999-2006 Landon Curt Noll ## ## Calc is open software; you can redistribute it and/or modify it under ## the terms of the version 2.1 of the GNU Lesser General Public License ## as published by the Free Software Foundation. ## ## Calc is distributed in the hope that it will be useful, but WITHOUT ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY ## or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General ## Public License for more details. ## ## A copy of version 2.1 of the GNU Lesser General Public License is ## distributed with calc under the filename COPYING-LGPL. You should have ## received a copy with calc; if not, write to Free Software Foundation, Inc. ## 59 Temple Place, Suite 330, Boston, MA 02111-1307, USA. ## ## @(#) $Revision: 29.4 $ ## @(#) $Id: ptest,v 29.4 2006/06/25 22:16:55 chongo Exp $ ## @(#) $Source: /usr/local/src/cmd/calc/help/RCS/ptest,v $ ## ## Under source code control: 1996/02/25 00:27:43 ## File existed as early as: 1996 ## ## chongo /\oo/\ http://www.isthe.com/chongo/ ## Share and enjoy! :-) http://www.isthe.com/chongo/tech/comp/calc/