Math::NumSeq::GoldbachCount -- number of representations as sum of primes P+Q


 use Math::NumSeq::GoldbachCount;
 my $seq = Math::NumSeq::GoldbachCount->new;
 my ($i, $value) = $seq->next;


The number of ways each i can be represented as a sum of two primes P+Q, starting from i=1,

    0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 2, 1, 2, 0, ...
    starting i=1

For example i=4 can be represented only as 2+2 so just 1 way. Or i=10 is 3+7 and 5+5 so 2 ways.

Even Numbers

Option on_values => 'even' gives the count on just the even numbers, starting i=1 for number of ways "2" can be expressed (none),

    0, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 3, 3, 2, 3, 2, 4, 4, ...
    starting i=1

Goldbach's famous conjecture is that for an even i >= 4 there's always at least one P+Q=i, which would be a count here always >= 1.

Odd Numbers

Odd numbers i are not particularly interesting. An odd number can only be i=2+Prime, so the count is simply

    count(odd i) = 1  if i-2 prime
                   0  if not


See "FUNCTIONS" in Math::NumSeq for behaviour common to all sequence classes.

$seq = Math::NumSeq::GoldbachCount->new ()
$seq = Math::NumSeq::GoldbachCount->new (on_values => 'even')

Create and return a new sequence object.

Random Access

$value = $seq->ith($i)

Return the sequence value at $i, being the number of ways $i can be represented as a sum of primes P+Q, or with the on_values=>'even' option the number of ways for 2*$i.

This requires checking all primes up to $i (or 2*$i) and the current code has a hard limit of 2**24 in the interests of not going into a near-infinite loop.

$bool = $seq->pred($value)

Return true if $value occurs as a count. All counts 0 upwards occur so this is simply integer $value >= 0.


Math::NumSeq, Math::NumSeq::Primes, Math::NumSeq::LemoineCount, Math::NumSeq::PrimeFactorCount



Copyright 2012, 2013, 2014, 2016, 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.

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.

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