20110714, 01:43  #1  
"Lucan"
Dec 2006
England
1100101001010_{2} Posts 
GPU TF work and its impact on P1
Quote:
On electric bill grounds, it makes NO sense to do LL tests on GPUs On all grounds, it makes no sense to do TF on CPUs. If any further disincentive were needed to explain the reluctance to embark on, and complete, a first time LL test, the thought that a GPU could TF an extra 6 bits in ~ a day is a further deterrent. This boosts the likelihood of primality by about 9%. (It may also render P1 pointless, but the clamour to do this in the 53M+ range is barely audible!) So GPUs contribute most effectively to the prime search by TFing an extra 6 bits in the 53M range. They should receive some acknowledgement (10%), should it yield a prime. David Last fiddled with by davieddy on 20110714 at 02:03 

20110714, 11:03  #2 
Dec 2010
Monticello
5·359 Posts 

20110714, 14:42  #3  
"Lucan"
Dec 2006
England
6474_{10} Posts 
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You know more than I do about the electricity needed to do a given LL test on a GPU instead of a CPU. LL for M4570xxxx 36% done. ETA Aug 12th. Celeron 440 @2GHz. Prime95 v25.11;( FFT 2560K. You already have the credit for TFing from 68 to 74 bits. When it turns out prime, I shall acknowledge your contribution and even pay for the electricity you used The probability of you finding no factor was 67/73, so the probability of one or more was 6/73 (~8%). BUT it had P1 done with B1=480000 B2=4080000. Can someone supply a simple formula to tell me what the probability of P1 finding a factor between 2^X and 2^(X+!) was? David Last fiddled with by davieddy on 20110714 at 15:02 

20110715, 22:24  #4 
Jun 2003
10010010001_{2} Posts 
mprime chose B1=530000, B2=13382500 for one of my assignments with the (correct) TF bit depth of 68. Changing the depth to 74, and leaving all other settings the same, it choose B1=355000, B2=5946250.
Extra TF makes P1 less valuable, which is reflected in the reduced bounds. But it does not make it pointless (assuming the program's calculation of the optimal bounds is correct, of course). 
20110715, 22:41  #5  
Jun 2003
7×167 Posts 
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20110715, 23:11  #6  
"Bob Silverman"
Nov 2003
North of Boston
2^{2}·1,877 Posts 
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The probability that a large integer N (N > oo) has a factor betwen x and x^(1+e) is e/(e+1). The proof I give uses Mertens' Theorem. 

20110716, 01:33  #7  
"William"
May 2003
Near Grandkid
2·1,187 Posts 
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20110716, 01:51  #8  
"Lucan"
Dec 2006
England
2×3×13×83 Posts 
Quote:
Probabity is 1/71. We knew this (except I thought it was 1/70). I am interested in the probability of P1 (with GIMPS optimized bounds) finding a factor between 2^70 and 2^71. From data helpfully supplied in This thread post#595 I judge that the probability of P1 finding a factor between 2^(70+x) and 2^(71+x) is P(x) = P(0)*2^(x/8) In the 53M exponent range, (well P1 ed) 1500 factors were found in about 21000 attempts (7% as roughly expected) I reckon P(0) ~ 7/12 % ~ 0.6% This tallies with George's recollection that P1 finds 3040% of the factors between 2^70 abd 2^71. David Last fiddled with by davieddy on 20110716 at 02:20 

20110716, 08:47  #9 
Jun 2003
7×167 Posts 
That statement is only an approximation, and it tells us about how the aggregate of P1 computations actually done by the project perform. It does not tell us how to relate the P1 bounds to the probability of success.

20110716, 10:57  #10  
"Lucan"
Dec 2006
England
2×3×13×83 Posts 
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supplied (see link) numbers of all known factors for exponents 4060M broken down by range and bits. The frequency for bits >72 halved every 8 bits, hence my guess that P(x) = (7/1200) 2^(x/8). Summing 2^(x/8) for x=0 to oo gives 12. So P1 finds 40% of 71 bit factors (for a "good" hit rate of 7%), hence George's 3040%. Each extra bit of TF reduces the yield of a subsequent P1 by 11/12. David BTW a plausible alternative formula is: P(x) = (7/1200) 2^(x/8) * 70/(70+x) But: 1) I can't readily sum it 2) The difference is unimportant for my intended purposes. Is there a theoretical justification for either version? 

20110716, 11:19  #11 
"Lucan"
Dec 2006
England
2×3×13×83 Posts 
@Mods:
Would posts from #16 onwards usefully be moved to "P1 factoring anyone?"? Last fiddled with by davieddy on 20110716 at 11:22 
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