++ed by:
Kevin Ryde
and 1 contributors

# NAME

Math::NumSeq::SophieGermainPrimes -- Sophie Germain primes p and 2*p+1 prime

# SYNOPSIS

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

# DESCRIPTION

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``````

# FUNCTIONS

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.)

# FORMULAS

## Next

`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?