Kevin Ryde > Math-NumSeq-63 > Math::NumSeq::ErdosSelfridgeClass

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Module Version: 63   Source   Latest Release: Math-NumSeq-69

# 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 on_values => '-' 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 => 'all')

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'.

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

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

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