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 2016-11-18, 15:35 #56 robert44444uk     Jun 2003 Oxford, UK 2·13·73 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 X-2(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.
2016-11-18, 16:38   #57
science_man_88

"Forget I exist"
Jul 2009
Dumbassville

202618 Posts

Quote:
 Originally Posted by robert44444uk 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 X-2(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.
my guess is it probably depends on gcd and a lot more to figure it out to an exact value for Y or Z.

2016-11-22, 11:43   #58
robert44444uk

Jun 2003
Oxford, UK

2·13·73 Posts

Quote:
 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 X-2(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.
I don't think the hypothesis I outlined above can be the case. The two values X and Y presumably share a large number of factors, given that P# and Q# are closely related, and hence the closest that X and Y would come would be defined by the multiple of their common factors.

Is this right?

Last fiddled with by robert44444uk on 2016-11-22 at 11:46

 2017-03-21, 17:15 #59 robert44444uk     Jun 2003 Oxford, UK 35528 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 2017-03-21 at 17:15
 2017-03-23, 18:23 #60 mart_r     Dec 2008 you know...around... 11278 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.
2017-03-23, 19:21   #61
danaj

"Dana Jacobsen"
Feb 2011
Bangkok, TH

38A16 Posts

Quote:
 Originally Posted by mart_r If you're looking for merit >30, D=30 is most effective for the P's you're looking at.
Looking at the top-20 records:
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

2017-03-23, 22:50   #62
robert44444uk

Jun 2003
Oxford, UK

111011010102 Posts

Quote:
 Originally Posted by mart_r 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.
Sorry Mart_r do you have your Ps and D's mixed up? The way I defined it, P is the prime in the primorial, D is the divisor and Y the multiplier.

2017-03-28, 18:49   #63
mart_r

Dec 2008
you know...around...

10010101112 Posts

Quote:
 Originally Posted by robert44444uk Sorry Mart_r do you have your Ps and D's mixed up? The way I defined it, P is the prime in the primorial, D is the divisor and Y the multiplier.
No, the P's and D's in my post are all where they have to be.
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.

 2017-07-19, 22:27 #64 ATH Einyen     Dec 2003 Denmark 23·7·53 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 "Maynard-Tao" 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) Maynard-Tao 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 2017-07-19 at 22:30
2017-07-20, 13:39   #65
CRGreathouse

Aug 2006

134548 Posts

Quote:
 Originally Posted by ATH I was curious about the value of this "Maynard-Tao" 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.
It's much larger than the Cramér-Shanks-Granville ratio -- which we expect to be bounded, or 'nearly' bounded, unlike the one you mention (Ford-Green-Konyagin-Tao).

The state of the art today is Ford-Green-Konyagin-Maynard-Tao:

C1 = Gn * (log log log Pn) / (log Pn * log log Pn * log log log log Pn).

 2017-07-20, 17:10 #66 robert44444uk     Jun 2003 Oxford, UK 35528 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 Now I have to get my head around what this means 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 2017-07-20 at 17:13

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