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 2018-05-20, 23:19 #23 Bobby Jacobs     May 2018 233 Posts 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)?
 2018-06-03, 23:28 #24 Bobby Jacobs     May 2018 E916 Posts 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
2018-06-04, 00:29   #25
science_man_88

"Forget I exist"
Jul 2009
Dumbassville

26·131 Posts

Quote:
 Originally Posted by Bobby Jacobs 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.

2018-06-04, 14:34   #26
ATH
Einyen

Dec 2003
Denmark

52×127 Posts

Quote:
 Originally Posted by Bobby Jacobs 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.

2018-06-04, 14:43   #27
CRGreathouse

Aug 2006

3×1,993 Posts

Quote:
 Originally Posted by Bobby Jacobs 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.

 2018-06-05, 22:58 #28 Bobby Jacobs     May 2018 111010012 Posts 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).
2018-06-06, 12:13   #29
CRGreathouse

Aug 2006

10111010110112 Posts

Quote:
 Originally Posted by Bobby Jacobs 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?

2018-06-06, 13:25   #30
Dr Sardonicus

Feb 2017
Nowhere

3×1,657 Posts

Quote:
 Originally Posted by Bobby Jacobs 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...

2018-06-07, 22:23   #31
Bobby Jacobs

May 2018

E916 Posts

Quote:
 Originally Posted by CRGreathouse 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.

 2018-06-17, 22:15 #32 Bobby Jacobs     May 2018 111010012 Posts 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
2018-06-18, 16:03   #33
Dr Sardonicus

Feb 2017
Nowhere

3·1,657 Posts

Quote:
 Originally Posted by Bobby Jacobs 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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