20161118, 15:35  #56 
Jun 2003
Oxford, UK
1,933 Posts 
A new idea
Let X = A*P#/(multiple of some primes)
We know that the interval X +/ 2P has very few values that might be prime. What is to stop us finding Y = B*Q#/(multiple of some other primes) that is close in value to X, in fact very close to or equal to X+2(P+Q) or X2(P+Q) Then we would have an interval approximately twice as long as I that would have very few values that might be prime, assuming P is similar in size to Q. By extension we could find Z=C#/(multiple of yet other primes) close to the range so that the interval is 3 times as long, etc. Simple algebra can find Y,Z I think, or am I missing something fundamental here. 
20161118, 16:38  #57  
"Forget I exist"
Jul 2009
Dumbassville
2^{6}×131 Posts 
Quote:


20161122, 11:43  #58  
Jun 2003
Oxford, UK
1,933 Posts 
Quote:
Is this right? Last fiddled with by robert44444uk on 20161122 at 11:46 

20170321, 17:15  #59 
Jun 2003
Oxford, UK
1,933 Posts 
I am wondering how we might categorise divisors, in order to find those that are most likely to provide records.
I'm looking at an approach using two measures, A  number of gaps >10 over a given range for a divisor B  "persistence"  the ratio of the number of gaps of size x+1 to the number of gaps of size x over a given test range for a divisor For a given divisor D, The test range I am using takes in multiple Y and P in Y*P#/D (centre points of large gaps). The specific ranges I am first looking at are D from 1 to 10,000 (squarefree only of course), Y from 1 to 5,000 and P from 97 to 229, so 130,000 tests for each D. I am not looking for all merit 10 gaps, as I am setting the delta =4. The probability of large gaps should be correlated to A and B, with B dominating, as explained below. The persistence measure is exponential. A 60% persistence suggests that 1 in 27,251 gaps of merit >10 will be a gap of merit 30 and 1 in 165 for a 20 merit gap. A 50% persistence on the other hand, 1 in every 1,048,576 gaps of merit >10 will actually be merit 30, and 1 in every 1,024 merit 20. So although the divisor 2 produces possibly the highest number of size 10 gaps (A=0.352%) its persistence ratio looks to be only in the 35% range, suggesting a conversion ratio of 1 in 1,314,132,370 to achieve a 30 merit gap and 1 in 36,251 for a merit 20. Other small Ds are showing >50% persistence, and A in excess of 0.2%. I know that my favoured divisor 46410 is closer to persistence ratio=60% with an A value relatively negligible. The plan is to find a divisor with persistence of 60% and a much higher A value. Grateful for your views. Last fiddled with by robert44444uk on 20170321 at 17:15 
20170323, 18:23  #60 
Dec 2008
you know...around...
1214_{8} Posts 
To get a persistence of more than 60%, the value for P must be higher. For P around 20,000 you get B>60% with D a primorial >= 17#, without having to cope with a small A value.
For P around 50,000, you can even choose, say, D=41#, and still get a decent A~4% (maybe more) with B>60~65%. Downside is, the tests take longer... If you're looking for merit >30, D=30 is most effective for the P's you're looking at. 
20170323, 19:21  #61  
"Dana Jacobsen"
Feb 2011
Bangkok, TH
2^{2}·227 Posts 
Quote:
7 D=30 7 D=210 2 D=2310 4 other (3 are maximal gaps, 1 is D=7230) Looking at allgaps.dat for merits >= 30, 476 D=30 427 D=210 56 D=2310 20 D=6 9 D=46410 This is heavily impacted by what's being searched for, but I believe D=30 has much more searching than other divisors. For all gaps >= 10 merits, D=30 has over 4 times as many results as the next (D=210). For gaps under 40k, percent of merits that are over 30. Again impacted by search ranges but maybe it tells us something: 7.17% D=30 7.54% D=210 7.99% D=2310 4.55% D=6 4.23% D=46410 27.78% D=9570 

20170323, 22:50  #62  
Jun 2003
Oxford, UK
1,933 Posts 
Quote:


20170328, 18:49  #63  
Dec 2008
you know...around...
1010001100_{2} Posts 
Quote:
Only my results may vary a bit more or less compared to yours. I'm basically looking at the count of coprimes mod P#. In my example, 50000#/41#, i.e. D=304250263527210, showed a persistance of >60%, according to your definition. 

20170719, 22:27  #64 
Einyen
Dec 2003
Denmark
110000111111_{2} Posts 
Regarding the "new" gap formula by Maynard, Tao and others
https://www.youtube.com/watch?v=BH1GMGDYndo https://arxiv.org/abs/1408.4505 G(X) >= C * log X * loglog X * loglogloglog X / (logloglog X)^2 They found out that C can get arbitrarily large as X>infinity. I was curious about the value of this "MaynardTao" constant for known gaps: C= Gn * (logloglog Pn)^2 / (log Pn * loglog Pn * loglogloglog Pn) It seems to follow the "Cramér–Shanks–Granville ratio" somewhat and is largest at the smaller gaps. Code:
Gn Merit Gn/(ln(Pn)^2) MaynardTao Pn 15900 39.62015365 0.09872683 36.37716367 1.936933265397289504398811903696*10^174 18306 38.06696007 0.07915948 34.09959273 7.041097148478282668812106731813*10^208 10716 36.85828850 0.12677617 35.50074155 1.839377720243795270729953508768*10^126 13692 36.59018324 0.09778276 33.93062260 3.254185929142547441117000456865*10^162 26892 36.42056789 0.04932537 30.92865059 4.696226774889053656642126142794*10^320 66520 35.42445941 0.01886489 27.27312299 3.292808201042179724620296543360*10^815 1476 35.31030807 0.84472754 59.57000119 1425172824437699411 1442 34.97568651 0.84833471 59.49933905 804212830686677669 1454 34.11893253 0.80062005 56.90602177 3219107182492871783 1370 33.76518602 0.83218087 58.00949419 418032645936712127 1132 32.28254764 0.92063859 61.34832684 1693182318746371 6582144 13.18288411 0.00002640 7.04128546 8.465069837806447347636518542879*10^216840 5103138 10.22031845 0.00002047 5.45890046 7.695421151871542659687327631743*10^216848 Last fiddled with by ATH on 20170719 at 22:30 
20170720, 13:39  #65  
Aug 2006
3^{2}×5×7×19 Posts 
Quote:
The state of the art today is FordGreenKonyaginMaynardTao: C1 = Gn * (log log log Pn) / (log Pn * log log Pn * log log log log Pn). 

20170720, 17:10  #66 
Jun 2003
Oxford, UK
1,933 Posts 
Using Antonio's examples and the FGKMT formula provides C as follows:
Code:
Gn C 15900 20.31243206 18306 18.72974166 10716 20.45427849 13692 19.07131193 26892 16.38392518 66520 13.501963 1476 45.22507308 1442 45.29864017 1454 43.03407437 1370 44.3097939 1132 48.34468117 6582144 2.735318749 5103138 2.12060946 I think we can say that we are nowhere near getting maximal gaps outside of the range we are searching suggesting that the merit is going to increase a lot. Last fiddled with by robert44444uk on 20170720 at 17:13 
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