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



Copyright 2010, 2011, 2012, 2013, 2014, 2016, 2017, 2019, 2020 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.

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