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Old 2018-05-20, 23:19   #23
Bobby Jacobs
 
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Here are some values of some prime gap measures. The first prime is p1, the second prime is p2, and the gap is g=p2-p1.

p1:
2, 3, 7, 23, 89, 113, 523, 887, 1129, 1327, 9551, 15683, 19609, 31397, 155921, 360653, 370261, 492113, 1349533, 1357201, 2010733, 4652353, 17051707, 20831323, 47326693, 122164747, 189695659, 191912783, 387096133, 436273009

g:
1, 2, 4, 6, 8, 14, 18, 20, 22, 34, 36, 44, 52, 72, 86, 96, 112, 114, 118, 132, 148, 154, 180, 210, 220, 222, 234, 248, 250, 282

p2:
3, 5, 11, 29, 97, 127, 541, 907, 1151, 1361, 9587, 15727, 19661, 31469, 156007, 360749, 370373, 492227, 1349651, 1357333, 2010881, 4652507, 17051887, 20831533, 47326913, 122164969, 189695893, 191913031, 387096383, 436273291

g/ln(p1):
1.442695041, 1.820478453, 2.055593370, 1.913573933, 1.782278478, 2.961466361, 2.875591620, 2.946443246, 3.129851463, 4.728345408, 3.928243586, 4.554708602, 5.261164230, 6.953520217, 7.192376566, 7.502537053, 8.735011648, 8.697998417, 8.359741402, 9.347822898, 10.19704418, 10.03068891, 10.80966760, 12.46145233, 12.44865987, 11.92209966, 12.27641993, 13.00297965, 12.64274696, 14.17528581

g/ln(p2):
0.9102392264, 1.242669869, 1.668129566, 1.781845226, 1.748744355, 2.890061790, 2.860130432, 2.936796132, 3.121281768, 4.711767984, 3.926631632, 4.553388044, 5.259754890, 6.951982323, 7.192044900, 7.502381009, 8.734805614, 8.697844701, 8.359689617, 9.347758521, 10.19699247, 10.03066729, 10.80966075, 12.46144487, 12.44865659, 11.92209850, 12.27641914, 13.00297877, 12.64274655, 14.17528535

g/ln2(p1):
2.081368981, 1.657070898, 1.056366025, 0.6102942000, 0.3970645718, 0.6264487862, 0.4593903979, 0.4340763900, 0.4452713720, 0.6575661849, 0.4286416021, 0.4714856920, 0.5323047896, 0.6715478251, 0.6015148915, 0.5863339816, 0.6812538258, 0.6636418989, 0.5922481044, 0.6619832798, 0.7025656074, 0.6533423380, 0.6491606315, 0.7394656865, 0.7044051476, 0.6402543257, 0.6440619075, 0.6817640317, 0.6393562030, 0.7125486806

g/ln2(p2):
0.8285354492, 0.7721142022, 0.6956640620, 0.5291620684, 0.3822633524, 0.5966040821, 0.4544636715, 0.4312385758, 0.4428363584, 0.6529634570, 0.4282898882, 0.4712123337, 0.5320196440, 0.6712508086, 0.6014594167, 0.5863095918, 0.6812216886, 0.6636184425, 0.5922407668, 0.6619741618, 0.7025584821, 0.6533395208, 0.6491598082, 0.7394648018, 0.7044047769, 0.6402542013, 0.6440618244, 0.6817639395, 0.6393561615, 0.7125486341

(g-ln2(p1))/ln(p1):
0.7495478602, 0.7218661633, 0.1096832205, -1.221920283, -2.706357891, -1.765921458, -3.383989844, -3.841401736, -3.899236100, -2.462330627, -5.236157553, -5.105624000, -4.622579692, -3.400947413, -4.764728184, -5.293134509, -4.086951795, -4.408465219, -5.755527761, -4.773112153, -4.316965716, -5.322194759, -5.842093269, -4.390516001, -5.223925161, -6.698781420, -6.784511621, -6.069571917, -7.131436667, -5.718492958

(g-ln2(p2))/ln(p2):
-0.1883730631, -0.3667680428, -0.7297657073, -1.585450605, -2.825966624, -1.954125295, -3.433288847, -3.873346319, -3.927104640, -2.504207017, -5.241531661, -5.109746216, -4.626637369, -3.404775885, -4.765611257, -5.293556689, -4.087460266, -4.408850569, -5.755666980, -4.773273778, -4.317091027, -5.322249480, -5.842110685, -4.390533533, -5.223933089, -6.698784388, -6.784513638, -6.069574086, -7.131437716, -5.718494068

The measure (g-ln2(p1))/ln(p1) does not have the same distribution for all maximal gaps because it is more negative for bigger numbers. However, if we have a better formula for G(p) than ln2(p), then it will have the same distribution. What is a better formula than ln2(p)?
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Old 2018-06-03, 23:28   #24
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Hi! Does anyone know a formula for the expected maximal prime gap for primes up to p?

Last fiddled with by Bobby Jacobs on 2018-06-03 at 23:29
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Old 2018-06-04, 00:29   #25
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Quote:
Originally Posted by Bobby Jacobs View Post
Hi! Does anyone know a formula for the expected maximal prime gap for primes up to p?
We know that p#+2 to p#+ nextprime after p -1 are all composite. But I don't know of a formula for maximal gap.
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Old 2018-06-04, 14:34   #26
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Quote:
Originally Posted by Bobby Jacobs View Post
Hi! Does anyone know a formula for the expected maximal prime gap for primes up to p?
https://arxiv.org/abs/1408.4505

https://arxiv.org/abs/1408.5110

Quote:
Let G(X) denote the size of the largest gap between consecutive primes below X. Answering a question of Erdos, we show that

G(X) \geq f(X) \frac{\log X \log \log X \log \log \log \log X}{(\log \log \log X)^2}

where f(X) is a function tending to infinity with X.


https://www.youtube.com/watch?v=BH1GMGDYndo&t=7m44s
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Old 2018-06-04, 14:43   #27
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Quote:
Originally Posted by Bobby Jacobs View Post
Hi! Does anyone know a formula for the expected maximal prime gap for primes up to p?
Bobby, I'm not sure that I understand your objection to the standard \log^2 p. Perhaps if you could explain in what way you feel this does not meet your needs it would help us find something more suitable for your situation.
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Old 2018-06-05, 22:58   #28
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If G(p) is the expected maximal prime gap between primes up to p, then (g-G(p))/ln(p) should be a measure with the same distribution on all maximal prime gaps. The measure (g-ln2(p))/ln(p) was not like that. There should be a better approximation than ln2(p).
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Old 2018-06-06, 12:13   #29
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Quote:
Originally Posted by Bobby Jacobs View Post
If G(p) is the expected maximal prime gap between primes up to p, then (g-G(p))/ln(p) should be a measure with the same distribution on all maximal prime gaps.
Why?
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Old 2018-06-06, 13:25   #30
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Quote:
Originally Posted by Bobby Jacobs View Post
If G(p) is the expected maximal prime gap between primes up to p, [snip]
"Expected" I understand. "Maximal" I understand. "Expected maximal," not so much...
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Old 2018-06-07, 22:23   #31
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Quote:
Originally Posted by CRGreathouse View Post
Why?
The standard deviation of the maximal gap seems to be about ln(p). Therefore, (g-G(p))/ln(p) is approximately the number of standard deviations above the expected maximal gap.
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Old 2018-06-17, 22:15   #32
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There is also something called the gap value. It is sqrt(g)/ln(p1+p2). Here are the gap values of the first 30 maximal prime gaps.

0.6213349347, 0.6800929643, 0.6919525124, 0.6199287913, 0.5412484188, 0.6827045970, 0.6087185235, 0.5969053332, 0.6066293050, 0.7384173974, 0.6085543782, 0.6405915434, 0.6816936284, 0.7679848950, 0.7330617929, 0.7263691872, 0.7830410800, 0.7737166488, 0.7335523374, 0.7755517622, 0.7999848732, 0.7733788478, 0.7735067688, 0.8259491127, 0.8076125106, 0.7714425767, 0.7743746631, 0.7967345566, 0.7725183220, 0.8157048503

This is similar to the CSG ratio because it approaches 1 with bigger maximal gaps.

Last fiddled with by Bobby Jacobs on 2018-06-17 at 22:18
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Old 2018-06-18, 16:03   #33
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Quote:
Originally Posted by Bobby Jacobs View Post
There is also something called the gap value. It is sqrt(g)/ln(p1+p2).
If you assume that g is small compared to p = p1, and

g=ln(p)k,

then the function value is roughly (ln(p))k/2 - 1.
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