Prime Gap Theory

[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.

[science_man_88]

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.

[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.

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?

[robert44444uk]

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.

[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.

[danaj]

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

[robert44444uk]

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.

[mart_r]

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.

[ATH]

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.

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

[CRGreathouse]

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).

[robert44444uk]

Using Antonio's examples and the FGKMT formula provides C as follows:

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.