%% Function File primes (n)
%% Return all primes up to 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 k primes less than k \log ( 5 k ).
%%
%% The algorithm used is called the Sieve of Erastothenes.
%% Author: Paul Kienzle, Francesco Potort
function x=primes(p)
if nargin ~= 1
error ('incorrect number of parameters');
end
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;
end % 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;
end
end
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
end
end
x = [2, 1+2*find(sieve)]; % primes remaining after sieve
else
x=[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, ...
61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, ...
131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193,...
197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269,...
271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349];
x = x (x<=p);
end
%% 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