20151016, 07:33  #34  
Jun 2003
Oxford, UK
1,933 Posts 
Quote:
Another one this morning: >338,000 gap, merit 21.7 

20151020, 11:50  #35 
Just call me Henry
"David"
Sep 2007
Cambridge (GMT/BST)
2×5×587 Posts 
Does getting high merits get harder when you get to larger gaps?

20151020, 16:06  #36  
"Dana Jacobsen"
Feb 2011
Bangkok, TH
2^{2}×227 Posts 
Quote:
Empirically in the 1008000 digit range, the BPSW test is about O(log^2.5(n)). 2x larger size is 56x longer time. The larger size also means a longer range for a large merit, which means more tests. Presumably log(n) growth. There is a complicating factor of the partial sieve that has a dynamic log^2(n) depth. Usually I see the tradeoff as small sizes run faster but are better covered hence need high merits to get a record. Large sizes (200k+) are slow but are so sparse that almost anything you find is a record. The sweet spot this year seems to be in the 7090k range for efficiency of generating records. There are lots of gaps with merit under 10. My stats page (http://ntheory.org/gaps/stats.pl) has a section that shows the gaps with merit < 20, < 15, < 10, and no gap. The strikethrough values are ones my unsubmitted gap set has found. I usually submit every 3 weeks or so. 

20151020, 17:50  #37 
Jun 2003
Oxford, UK
3615_{8} Posts 
I barely have a positive score for this week, looking at the number of my records that Danaj has clawed back
The good news is that the new ones are a better quality  almost everything I am looking for is 150k plus so should last a while. Last fiddled with by robert44444uk on 20151020 at 17:51 
20151020, 21:07  #38 
Just call me Henry
"David"
Sep 2007
Cambridge (GMT/BST)
2·5·587 Posts 
So basically the merit isn't a good measure of how hard it is to find them. Maybe we should correct the formula.

20151020, 21:52  #39  
Dec 2008
you know...around...
2·5^{2}·13 Posts 
Quote:
This gives sort of an "effective" merit, as displayed in this example: Code:
center number = 2000003# / 13# numbers without factor <= 2000003 effective merit  side + side  side + side merit ± 1 2550 2527 0.03 0.03 merit ± 2 3218 3199 0.04 0.04 merit ± 3 21172 21119 0.27 0.27 merit ± 4 38603 38594 0.50 0.50 merit ± 5 64610 64486 0.84 0.83 merit ± 6 90082 90090 1.16 1.16 merit ± 7 127014 127067 1.64 1.64 merit ± 8 163654 163684 2.12 2.12 merit ± 9 204374 204397 2.64 2.64 merit ±10 244814 244884 3.17 3.17 If e.g. you aim for a merit >10 in this region (± 5), after four attempts there is a >50% chance that an example is found. (I loosely calculate this 50%chance by using the factor log(2), so exp(0.84+0.83)*log(2) ~ 3.7 attempts) 

20151020, 22:39  #40 
"Dana Jacobsen"
Feb 2011
Bangkok, TH
2^{2}·227 Posts 
A little experiment looking at the time and number of merits >= 5.0 found using k*p#/30b where k=1..10000 without multiples of 2,3,5.
p=20: 1.7s 102 found = 60/s (2830 digits) p=40: 4.1s 236 found = 58/s (6971 digits) p=80: 19.6s 515 found = 26/s (166169 digits) p=160 235s 985 found = 4/s (392395 digits) Interestingly with this form the number we find with merit >= 5 goes up as we get larger. But the time taken goes up quite a bit faster. Which would explain why we see the shape of the graph of current records (high at the beginning, dropping off as gap size increases). As discussed on the other prime gap thread, it's certainly possible that a different method of selecting the search points would be more efficient, and it's also possible to improve the speed of this or other methods vs. doing prev/next prime with my GMP code. For example with numbers larger than ~3000 digits using gwnum would be faster than GMP. gapcoin uses a different method, but it's not obvious to me how to get exact efficiency comparisons. From Mathworld: "The merit of a prime gap compares the size of a gap to the local average gap [...]" which doesn't take into account computational complexity nor the state of current records. 
20151021, 07:02  #41  
Jun 2003
Oxford, UK
1,933 Posts 
Quote:
Gaps can be arbitrarily long, so a proportionately large gap size is what makes a gap rare and there is a limit to gap size for a given set of two consecutive primes. Setting higher limits to reporting them, such as a merit of >10 does at least provide some discrimination. At any size of prime, beyond the 6,30, 210, 2310 rule, in general the larger the gap the rarer it is and the form of number we are searching  near the primorials  does lend itself to larger gaps  however, we are way off getting to a gap merit of 37. I live in hope. 

20151021, 07:58  #42 
Romulan Interpreter
Jun 2011
Thailand
2^{5}×5×59 Posts 
Fixed it for you.
Joking apart (or not), the last discussion made me think to analogy with the "difficulty" concept at bitcoin poolmining, where there is a "bound" for reporting. For example, if you do mining in a pool, the pool will pay you according with your efforts, and the only way it can check how much effort you did is if you report your results which are "better than the bound". So, the pool counts how many results like that are sent by any participant, and they constitute "shares" earned by the participant. The pool gets rewarded (paid) by the bitcoin community when one participant is enough lucky to find a result which is better than the (higher bound) "difficulty" set by the bitcoin community, and in that case the benefit (money) is shared with all participants in the pool according with their "shares". Next step here is to make a "gapcoin". The there will be a lot of participants.... Last fiddled with by LaurV on 20151021 at 08:02 
20151021, 16:01  #43 
"Dana Jacobsen"
Feb 2011
Bangkok, TH
2^{2}×227 Posts 
(apologies if I missed the sarcasm/humor tag)
There is such a thing, and it's called gapcoin. Forum / Site. They even have a GPU miner. Over the last year (today is their 1 year anniversary) it has found 1517 record gaps. Antonio has found more than that using a single computer over 3 months. We don't know what resources are being used for Gapcoin. However, Gapcoin is searching for smaller gaps, and has 2 top20overall records. The participants also get a coin, where we don't. You could make an argument that they're looking for higher quality gaps. For people with >100 gaps, average merit: 27.3 Nyman 26.8 Gapcoin 26.2 Spielaur 25.4 Gapcoin unsubmitted 18.7 Jacobsen 18.7 MJPC&JKA 17.9 Jacobsen unsubmitted 17.4 Brent 16.4 TonyKey 15.7 Jansen 15.4 RobSmith 14.9 Rosnthal 14.1 Cami 13.8 Andersen 13.1 TorAlmJA 
20151022, 02:29  #44 
Romulan Interpreter
Jun 2011
Thailand
2^{5}×5×59 Posts 
It wasn't sarcasm. I use different icons for that
Now I am thinking it was a bit silly for me not to google it before posting, but robert44's talk about "setting higher limits to reporting them" was crying in my brain to a similarity with pool mining, from which it was a very small step to where the name struck. Now, it seems just normal that someone else was thinking to it before me... Tragedy of my life... OTOH, I like their page (the one you linked). Good and elementary explanations for both the coining system and prime gaps' math, for everybody to understand. As an "old salt" bitcoin folk, I will give their GPU miner a try during this long weekend we will have here (23rd is national holiday). To see how fast it is. I have no real feeling (beside of this thread) what the numbers you posted really mean, in term of effort. Seeing you on the list, it may be not easy to get some coins ... Edit2: "there is no pool mining at present". Time to make one? Last fiddled with by LaurV on 20151022 at 02:42 
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