After n=333333, our next exponent will be... (revised poll)

[Skligmund]

I'd like to change my vote to 460-520.

[jmblazek]

As for where the twin and single primes will be in the Top 5000 list (as of 29 Jan 07):

 
n=   333,333  Rank in list 3003
n=   400,000  Rank in list 1415
n=   500,000  Rank in list  820
n=   666,666  Rank in list  386
n=   750,000  Rank in list  257
n= 1,000,000  Rank in list  150
n= 1,500,000  Rank in list   54
n= 2,000,000  Rank in list   32
n= 3,000,000  Rank in list   13
n= 5,000,000  Rank in list   11
n=10,000,000  Rank in list    7
n=25,000,000  Rank in list    4
n=33,000,000  Rank in list    1

Here's a collection of information from this thread and others.

A proposed roadmap:

Goal :   Digits :    n :     Remarks :
------- --------- ---------- ---------
1     58,710        195,000
2    100,000        332,190
3    175,258        582,192 > 500,000 exponent*
4    200,000        664,383
5    400,515      1,330,480 >1,000,000 exponent*
6    500,000      1,660,961
7    801,030      2,660,962 >2,000,000 exponent*
8  1,000,000      3,321,925
9  3,252,575     10,804,819 >5,000,000 exponent*
10 5,000,000     16,609,638

Computing effort needed for n: (from biwema)

Benchmark Computer: P4 3.6 GHz (from benchmark page)

Sieving information:

How long will it take to sieve? The sieving speed does not depend on the size of the range. Nevertheless, the size of the array is limited by the memory. The increasing size could make it difficult to merge over the web.
This list shows the sieving time (sieving speed 100 Million/ second) and the number of twin candidates per G.

bits | candidates | sieving time    
40     545700       3 hours
45     431170       3.7 days
50     349248       3.8 months
55     288634      10 years
60     242533     320 years
65     206655      10 K years
70     178187     330 K years
75     155221      10 million years
80     136425     320 million years
85     120847      10 billion years

LLR/Proth test effort.

n=172500
FFT size: 16K
Time 1 test: 51.75 s
Optimal sieve bits: 51
Tests per twin: 3.73M
Twin every G: 10.69
90% chance after G: 24.6
CPU years per twin: 5.9
 
n=195000
FFT size: 20K
Time 1 test: 72.15 s
Optimal sieve bits: 52
Tests per twin: 4.41M
Twin every G: 13.6
90% chance after G: 31.4
CPU years per twin: 10.1
 
n=250000
FFT size: 24K
Time 1 test: 115 s
Optimal sieve bits: 53
Tests per twin: 7M
Twin every G: 22.45
90% chance after G: 51.7
CPU years per twin: 25.4
 
n=333333
FFT size: 32K
Time 1 test: 200 s
Optimal sieve bits: 55
Tests per twin: 11.5M
Twin every G: 39.9
90% chance after G: 91.9
CPU years per twin: 73
 
n=500000
FFT size: 48K
Time 1 test: 500 s
Optimal sieve bits: 57.5
Tests per twin: 23.7M
Twin every G: 90
90% chance after G: 207
CPU years per twin: 376
 
n=1000000
FFT size: 112K
Time 1 test: 2800 s
Optimal sieve bits: 62
Tests per twin: 82M
Twin every G: 359
90% chance after G: 827
CPU years per twin: 7243
 
n=3330000
FFT size: 384K
Time 1 test: 30303
Optimal sieve bits: 69
Tests per twin: 730M
Twin every G: 3983
90% chance after G: 9170
CPU years per twin: 702000
 
n=33300000
FFT size: 3584K
Time 1 test: 3506490
Optimal sieve bits: 82
Tests per twin: 51.7G
Twin every G: 398300
90% chance after G: 917000
CPU years per twin: 5750 M

Conclusions:
- Optimal sieving goes deeper than expected.
- Sieving 1Mdigit twins requires 32G Ram (At the beginning more)
- Time complexity: 10 times more digits requires more than 10000 times more processing time.
- Sieving memory complexity: 10 times more digits requires 100 times more space.

[MooMoo2]

The poll has closed, and the n after n=333,333 will be n=500,000.