## Copyright (C) 2000 Paul Kienzle ## ## This program is free software; you can redistribute it and/or modify ## it under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 2 of the License, or ## (at your option) any later version. ## ## This program 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 General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with this program; if not, write to the Free Software ## Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA ## -*- texinfo -*- ## @deftypefn {Function File} {} primes (@var{n}) ## Return all primes up to @var{n}. ## ## Note that if you need a specific number of primes, you can use the ## fact the distance from one prime to the next is on average ## proportional to the logarithm of the prime. Integrating, you find ## that there are about @math{k} primes less than @math{k \log ( 5 k )}. ## ## The algorithm used is called the Sieve of Erastothenes. ## @end deftypefn ## Author: Paul Kienzle, Francesco Potort́ and Dirk Laurie function x=primes(p) if nargin != 1 usage("p = primes(n)"); endif if (p > 100000) ## optimization: 1/6 less memory, and much faster (asymptotically) ## 100000 happens to be the cross-over point for Paul's machine; ## below this the more direct code below is faster. At the limit ## of memory in Paul's machine, this saves .7 seconds out of 7 for ## p=3e6. Hardly worthwhile, but Dirk reports better numbers. lenm = floor((p+1)/6); # length of the 6n-1 sieve lenp = floor((p-1)/6); # length of the 6n+1 sieve sievem = ones (1, lenm); # assume every number of form 6n-1 is prime sievep = ones (1, lenp); # assume every number of form 6n+1 is prime for i=1:(sqrt(p)+1)/6 # check up to sqrt(p) if (sievem(i)) # if i is prime, eliminate multiples of i sievem(7*i-1:6*i-1:lenm) = 0; sievep(5*i-1:6*i-1:lenp) = 0; endif # if i is prime, eliminate multiples of i if (sievep(i)) sievep(7*i+1:6*i+1:lenp) = 0; sievem(5*i+1:6*i+1:lenm) = 0; endif endfor x = sort([2, 3, 6*find(sievem)-1, 6*find(sievep)+1]); elseif (p > 352) # nothing magical about 352; just has to be greater than 2 len = floor((p-1)/2); # length of the sieve sieve = ones (1, len); # assume every odd number is prime for i=1:(sqrt(p)-1)/2 # check up to sqrt(p) if (sieve(i)) # if i is prime, eliminate multiples of i sieve(3*i+1:2*i+1:len) = 0; # do it endif endfor x = ; # primes remaining after sieve else a=; x = x (x<=p); endif endfunction

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