## Combination sieving was Re: Sophie Germain record

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• ... Sieving combinations means a single sieve which checks multiple conditions. You get a single output file. Examples Twin/SG sieve - sieve k.2^n-1, k.2^n+1,
Message 1 of 2 , Feb 5, 2003
At 11:26 PM 2/4/2003 -0000, you wrote:
>> 1. Sieve combinations, not for a single SG. You will notice I did
>not have
>> to flood the list with thousands of single primes before finding
>the twin
>> pair and the SG pair.
>
>What do you do with the three sieved text files after you're done
>sieving? Is there a program that can sieve between the files to
>eliminate non-SG possible primes?
>
>dave k
>
>

Sieving combinations means a single sieve which checks multiple conditions.
You get a single output file. Examples

Twin/SG sieve - sieve k.2^n-1, k.2^n+1, and k.2^(n+1)-1. Possible pairs are
(k.2^n-1, k.2^n+1) and twin pair or
(k.2^n-1, k.2^(n+1)-1) and Sophie Germain pair.

What NewPGen calls a Lucky Minus sieve - sieve k.2^n-1, k.2^n+1,
k.2^(n+1)-1, and k.2^(n-1)-1. You have a possibility of a twin pair and two
SG pairs.

PRP test only the common candidate, which is k.2^n-1 it these examples. You
have to edit the header line in the output file for PRP to run only the
single test you desire. For each prime found PRP test the other conditions.
You are 2-4 times more likely to find a prime pair than if you ran only a
twin sieve or only a SG sieve.

First some data, then more details

Consider the example of sieving k.1999# up to p=100G and then running a PRP
test on the first 50,000 candidates in the output file. Here are the
examples from 3 actual sieves.

Sieve type PRPs found Twins Found SG found Total pairs
Twin 1052 29 0 29
Twin and SG 1147 35 25 60
Lucky Minus 1182 33 76 109

For a 50,000 candidate PRP run, a twin sieve found 29 pairs, twin/SG found
60, and LM found 109. I had similar results with a k.2^n form rather than a
primorial form except the LM test was not quite as beneficial.

The down side is that the sieve produces many less candidates in this mode
so you have to sieve as wide as possible. I seived a k range of 1 to 4
billion. The results can still be processed in PRP by this range. I also
sieve even and odd K values to double the number of candidates produced by
the sieve.

In the record SG effort I actually used a CC1 (Cunningham Chain 1st Kind)
sieve since I was not interested in a twin prime at that time.