# Copyright 2010, 2011, 2012, 2013, 2014, 2016, 2019, 2020 Kevin Ryde
# This file is part of Math-NumSeq.
#
# 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/>.
package Math::NumSeq::Primes;
use 5.004;
use strict;
use POSIX ();
use Math::Prime::XS 0.23 'is_prime'; # version 0.23 fix for 1928099
use vars '$VERSION', '@ISA';
$VERSION = 75;
use Math::NumSeq;
@ISA = ('Math::NumSeq');
*_is_infinite = \&Math::NumSeq::_is_infinite;
# uncomment this to run the ### lines
# use Smart::Comments;
# use constant name => Math::NumSeq::__('Prime Numbers');
use constant description => Math::NumSeq::__('The prime numbers 2, 3, 5, 7, 11, 13, 17, etc.');
use constant characteristic_increasing => 1;
use constant characteristic_integer => 1;
use constant values_min => 2;
use constant i_start => 1;
#------------------------------------------------------------------------------
# cf A010051 - characteristic boolean 0 or 1 according as N is prime
# A051006 characteristic as binary fraction, in decimal
# A051007 characteristic as binary fraction, continued fraction
# A000720 - pi(n) num primes <= n
# A018252 - the non-primes
# A002476 - primes 3k+1, which is also 6k+1
#
use constant oeis_anum => 'A000040'; # primes
#------------------------------------------------------------------------------
use constant 1.02; # for leading underscore
use constant _MAX_PRIME_XS => do {
my $umax = POSIX::UINT_MAX() / 2;
# if ($umax > 0x8000_0000) {
# $umax = 0x8000_0000;
# }
$umax;
};
sub rewind {
my ($self) = @_;
$self->{'i'} = $self->i_start;
$self->{'array_lo'} = 1;
$self->{'array_hi'} = 1;
@{$self->{'array'}} = ();
}
# needs a prime_count() for arbitrary seek
#
# sub _UNTESTED__seek_to_value {
# my ($self, $value) = @_;
# my $array = $self->{'array'};
# if (@array) {
# if ($value >= $array->[0] && $value <= $array->[-1]) {
# # seek forward within $array
# while ($value > $array->[0]) {
# shift @$array;
# $self->{'i'}++;
# }
# return;
# }
# }
# $value = int($value);
# if ($value > _MAX_PRIME_XS) {
# # past limit
# $self->{'array'} = undef;
# return;
# }
# $self->{'i'} = _primes_count(0,$value);
# $self->{'array_lo'} = 0;
# $self->{'array_hi'} = $value-1;
# @{$self->{'array'}} = ();
# }
sub next {
my ($self) = @_;
while (! @{$self->{'array'}}) {
# fill array
my $lo = $self->{'array_lo'};
my $hi = $self->{'array_hi'};
$lo = $self->{'array_lo'} = $hi+1;
if ($lo > _MAX_PRIME_XS) {
return;
}
my $len = int ($lo / 2);
if ($len > 100_000) {
$len = 100_000;
}
$hi = $lo + $len;
if ($hi < 500) {
$hi = 500;
}
if ($hi > _MAX_PRIME_XS) {
$hi = _MAX_PRIME_XS;
}
$self->{'array_hi'} = $hi;
@{$self->{'array'}} = _primes_list ($lo, $hi);
}
return ($self->{'i'}++, shift @{$self->{'array'}});
}
sub _primes_list {
my ($lo, $hi) = @_;
### _my_primes_list: "$lo to $hi"
if ($lo < 0) {
$lo = 0;
}
if ($hi > _MAX_PRIME_XS) {
$hi = _MAX_PRIME_XS;
}
if ($hi < $lo) {
# Math::Prime::XS errors out if hi<lo
return;
}
return Math::Prime::XS::sieve_primes ($lo, $hi);
}
sub pred {
my ($self, $value) = @_;
### pred(): "$value"
if (_is_infinite($value) || $value > 0xFFFF_FFFF) {
return undef;
}
if ($value != int($value) || $value < 0) {
return 0;
}
return is_prime($value);
}
# sub ith {
# my ($self, $i) = @_;
# my $array = $self->{'array'};
# if ($i > $#$array) {
# my $hi = int ($i/log($i) * 2 + 5);
# do {
# $array = $self->{'array'} = [ undef, _my_primes_list (0, $hi) ];
# $hi *= 2;
# } while ($i > $#$array);
# }
# return $array->[$i];
# }
use Math::NumSeq::Fibonacci;
*_blog2_estimate = \&Math::NumSeq::Fibonacci::_blog2_estimate;
sub value_to_i_estimate {
my ($self, $value) = @_;
### value_to_i_estimate(): "$value"
if ($value < 2) { return 0; }
$value = int($value);
if (defined (my $blog2 = _blog2_estimate($value))) {
# est = v/log(v)
# log2(v) = log(v)/log(2)
# est = v/((log2(v)*log(2)))
# = v/log2(v) * 1/log(2)
# ~= v/log2(v) * 13/9
# ~= (13*v) / (9*log2(v))
# using 13/9 as an approximation to 1/log(2) to stay in BigInt
#
### $blog2
### num: $value*13
### den: 9 * $blog2
return ($value * 13) / (9 * $blog2);
}
### log: log($value)
### div: $value/log($value)
return int($value/log($value));
}
1;
__END__
=for stopwords Ryde Math-NumSeq
=head1 NAME
Math::NumSeq::Primes -- prime numbers
=head1 SYNOPSIS
use Math::NumSeq::Primes;
my $seq = Math::NumSeq::Primes->new;
my ($i, $value) = $seq->next;
=head1 DESCRIPTION
The prime numbers, not divisible by anything except themselves and 1.
2, 3, 5, 7, 11, 13, 17, 19, ... (A000040)
starting i=1
Currently this is implemented with C<Math::Prime::XS> generating blocks of
primes with a sieve of Eratosthenes. The result is reasonably progressive.
On a 32-bit system there's a hard limit at 2^31 (though even approaching
that takes a long time to calculate).
=head1 FUNCTIONS
See L<Math::NumSeq/FUNCTIONS> for behaviour common to all sequence classes.
=over 4
=item C<$seq = Math::NumSeq::Primes-E<gt>new ()>
Create and return a new sequence object.
=item C<$bool = $seq-E<gt>pred($value)>
Return true if C<$value> is a prime.
In the current code a hard limit of 2**32 is placed on the C<$value> to be
checked, in the interests of not going into a near-infinite loop.
=item C<$i = $seq-E<gt>value_to_i_estimate($value)>
Return an estimate of the i corresponding to C<$value>. C<$value> can be
any size, it's not limited as in C<pred()>. See L</Value to i Estimate>
below.
=back
=head1 FORMULAS
=head2 Value to i Estimate
In the current code the number of count of primes up to value is estimated
by the well-known asymptotic
i ~= value/log(value)
=head1 SEE ALSO
L<Math::NumSeq>,
L<Math::NumSeq::TwinPrimes>,
L<Math::NumSeq::SophieGermainPrimes>,
L<Math::NumSeq::Emirps>
L<Math::Prime::XS>,
L<Math::Prime::TiedArray>,
L<Math::Prime::FastSieve>,
L<Math::Prime::Util>
=head1 HOME PAGE
L<http://user42.tuxfamily.org/math-numseq/index.html>
=head1 LICENSE
Copyright 2010, 2011, 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.
You should have received a copy of the GNU General Public License along with
Math-NumSeq. If not, see <http://www.gnu.org/licenses/>.
=cut