Math::NumSeq::SophieGermainPrimes -- Sophie Germain primes p and 2*p+1 prime
use Math::NumSeq::SophieGermainPrimes; my $seq = Math::NumSeq::SophieGermainPrimes->new; my ($i, $value) = $seq->next;
The primes P for which 2*P+1 is also prime,
2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113, 131, 173, 179, ... starting i=1
See "FUNCTIONS" in Math::NumSeq for behaviour common to all sequence classes.
$seq = Math::NumSeq::SophieGermainPrimes->new ()
Create and return a new sequence object.
$bool = $seq->pred($value)
Return true if $value
is a Sophie Germain prime, meaning both $value
and 2*$value+1
are prime.
$i = $seq->value_to_i_estimate($value)
Return an estimate of the i corresponding to $value
.
Currently this is the same as the TwinPrimes estimate. Is it a conjecture by Hardy and Littlewood that the two are asymptotically the same? In any case the result is roughly a factor 0.9 too small for the small to medium size integers this module might calculate. (See Math::NumSeq::TwinPrimes.)
next()
is implemented by a Math::NumSeq::Primes
sequence filtered for primes where 2P+1 is a prime too. Dana Jacobsen noticed this is faster than running a second Primes iterator for primes 2P+1. This is since for a prime P often 2P+1 has a small factor such as 3, 5 or 11. A factor 3 occurs for any P=6k+1 since in that case 2P+1 is a multiple of 3. What else can be said about the density or chance of a small factor?
Math::NumSeq, Math::NumSeq::Primes, Math::NumSeq::TwinPrimes
http://user42.tuxfamily.org/math-numseq/index.html
Copyright 2010, 2011, 2012, 2013, 2014 Kevin Ryde
Math-NumSeq 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 3, or (at your option) any later version.
Math-NumSeq 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 Math-NumSeq. If not, see <http://www.gnu.org/licenses/>.