In-Depth Analyses

65536

Note - this was written before Yves extended teh range of testable candidates, and is out of date. It applies to all 'small' candidates though

No in-depth analysis, but I do have the following testimonial:

 I once tried to sieve the range I received. Your list
 has 644 numbers for the 4000 range. I sieved the range
 down to 700 numbers in less than a week. Then I found
 out the rate was about 4 to 6 each day but I could test
 15 per day. So I quited sieving and went ahead with
 your list.

131072

Note - this was written before Yves expanded the useful range. The conclusions aren't vastly different now though. TrialDiv can be taken much further as PRP testing takes much longer.

I am sieving an overly broad range "unnecessarily", by a factor of 19.7, assuming that useful means the unreserved ranges as of 3rd Nov 2002,
58000-530000 + 626000-1166000 = 1012000
From here on, "useful" implies within that range.

How many candidates can be removed

Historical data - a snapshot of behaviour between 40Q and 50Q

count  P/Q product
2128   40  85120
2024   41  82984
1978   42  83076
1932   43  83076
1885   44  82940
1907   45  85815
1796   46  82616
1776   47  83472
1789   48  85872
1674   49  82026

True-sieving

A quintillion takes 4.5 days/GHz; i.e. I sieve at .22Q/(GHz.day)

=> at 70Q that's 60 useful candidates per extra Q.
=> at 70Q that's 60 per 4.5 days, or 13.4/day or 107min/GHz.

True-sieve cost = 107m

Trial-division sieving

Taking 100% of the useful range into trial-division mode, effectively trying to do the whole useful task that I'm doing presently with the true sieve, I could theoretically achieve 0.009Q/(GHz.day). This is 24 times slower than true sieving, coming to 44 hours/candidate.

Taking 1% of the useful range into trial-division mode, I can achieve .85Q/(GHz.day). In a day I'd expect to remove 0.85*0.01*60 = .51 candidates. 0.51/(GHz.day) is 2 days per candidate!

Taking 0.1% of the useful range, which is about the same size range that the single-PC contributors will be, I can achieve 6.7Q/(GHz.day). In a day I'd expect to remove 6.7*0.001*(60+55)/2 = 0.39 candidates. That's one every 62 hours.

Trial Division (block) cost = 44h Trial Division (solo) cost = 48-62h

Proth Testing

b=1168000 seems to takes 16.5hours/GHz.

Conclusions

1. Sieving should continue, using the true sieve.
2. There is no point in testers doing any further trial division.
3. There is NO conclusion 3.