Kevin Ryde > Math-NumSeq-71 > Math::NumSeq::ErdosSelfridgeClass

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NAME ^

Math::NumSeq::ErdosSelfridgeClass -- Erdos-Selfridge classification of primes

SYNOPSIS ^

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

DESCRIPTION ^

This is a class number for primes by Erdos and Selfridge, or 0 for composites. The default is "class+"

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

A prime p is classified by factorizing p+1 into primes, then on each of those primes q factorizing q+1, and so on, repeating until reaching entirely 2s and 3s. p=2 or p=3 interchange on factorizing p+1 (2+1=3 and 3+1=2*2).

A prime p where p+1 factorizes to all 2s or 3s is class 1. For example i=11 has 11+1=12=2*2*3 which is all 2s and 3s so class 1. 2 and 3 themselves are class 1 too, since their p+1 factorizing gives 2s and 3s.

Further primes are classified by how many iterations of the p+1 factorizing is necessary to reach 2s and 3s. For example prime p=3847 is iterated as

    3847+1 = 2*13*37

    then 13+1 = 2*7
         37+1 = 2*19

    then 7+1 = 2*2*2
         19+1 = 2*2*5

    then 5+1 = 2*3

So 3847 is class 4 as it took 4 steps to reach all 2s and 3s. Some of the factors become 2s and 3s earlier, but the steps continue until all factors are reduced to 2s and 3s.

Class -

Option p_or_m => '-' applies the same procedure to prime factors of p-1, giving a "class-" number.

    0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, ...

It sometimes happens that class+ is the same as class-, but in general the two are unrelated.

Primes Only

Option on_values => 'primes' selects the classes of just the primes,

    1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 2, 2, 1, 1, 2, 2, 2, 1, 4, ...

FUNCTIONS ^

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

$seq = Math::NumSeq::ErdosSelfridgeClass->new ()
$seq = Math::NumSeq::ErdosSelfridgeClass->new (p_or_m => $str, on_values => $str)

Create and return a new sequence object.

p_or_m (a string) can be

    "+"    factors of p+1 (the default)
    "-"    factors of p-1

on_values (a string) can be

    "all"      classify all integers
    "primes"   classify just the primes

Random Access

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

Return the class number of $value, or 0 if $value is not a prime.

This method is only available for the default on_values=>'all'. $seq->can('ith') says whether ith() can be used (and gives a coderef).

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

Return true if $value occurs as a classification, which means any integer $value >= 0.

SEE ALSO ^

Math::NumSeq, Math::NumSeq::Primes

HOME PAGE ^

http://user42.tuxfamily.org/math-numseq/index.html

LICENSE ^

Copyright 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/>.

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