Long ranges of gaps larger than a given value

robert44444uk · 2021-12-02, 18:17

Take a range of integers p1 to p(n+1) within which there are primes p1,p2,p3...p(n+1), with gaps between these primes of g1,g2,g3...gn

Take an integer to represent the smallest gap in the set g1,g2,g3..gn. Let this smallest gap be x

What is the smallest p1 for which x=2,4,6,8,... for each n?

If x=4, n=3, then the p1 is 43, as 43,47,53,59 is the first instance of three gaps 4,6,6 that are of size 4 or more

The table below shows values at n 1..20 in column 1, then for various x in the other columns. The table values are p1.

For x = 1,2 the p1 are 2 and 3 respectively.

For clarity I also attach a photo of the table.

Sorry if this duplicates other's work, but I have not seen any lists like this in OEIS.

Code:

	4	6	8	10	12	14	16	18	20	30	50	100	200
1	7	23	89	113	113	113	523	523	887	1327	19609	370261	20831323
2	19	47	199	199	199	1831	2161	2161	3947	24251	413299	72546143	15318488071
3	43	241	683	773	3947	7351	7351	7351	18593	69557	3021173	1284352451	
4	73	241	683	7043	10181	23371	63913	66191	112403	668243	14784911	24964088161	
5	73	523	683	13477	23087	47161	68281	68281	250501	2352289	107282507	82504446809	
6	349	677	1789	13477	23371	47161	231109	231109	783877	4720943	837406403	438290428309	
7	349	677	13469	13477	23371	339341	339341	539899	2352247	30097883	2547259079		
8	349	677	13469	13477	23371	339841	339841	539899	2722561	64740587	7623725903		
9	349	2861	13469	13477	23371	611561	876853	5556769	5556769	130925771	47787139219		
10	349	10733	13469	115153	257503	611561	6463309	6463309	7063817	1075978859	96622630009		
11	349	10733	62687	202409	406883	611561	7063817	7063817	7063817	1415929063	177013585123		
12	661	13421	62687	303731	406883	876853	7591729	12199571	61501877	2695118249	1324527983897		
13	661	13421	62687	303731	729979	876853	15022753	15022753	148486951	3731002741			
14	661	13421	62687	406817	729979	9355501	15022753	15022753	254259631	7780262561			
15	661	13421	200597	406817	729979	12877433	20917291	20917291	562725353	33705791429			
16	661	13421	200597	406817	729979	12877433	72733109	89085481	1143902897	46318910903			
17	661	13421	568441	913639	4656469	12877433	134798647	180820951	1462517893	103947089669			
18	661	13421	568441	913639	4656469	12877433	195503851	195503851	2158109033	103947089669			
19	661	13421	568441	970447	4656469	12877433	232253071	381906059	4527970103	184722051989			
20	8629	13421	568441	970447	8962013	112843903	232253071	381906059	9156493673	184722051989

robert44444uk · 2021-12-03, 12:06

The first instance of 7 consecutive 100+ gaps is 5183073661943 with gaps 116,130,114,158,286,102,102

The first instance of 3 consecutive 200+ gaps is provided by 20222645954633 with gaps of 204,216,218

robert44444uk · 2021-12-04, 08:10

The first instance of 13 and 14 consecutive 50+ gaps is at 3057601284499 where subsequent gaps are 60, 54, 58, 50, 100, 90, 62, 156, 124, 60, 60, 68, 76 and 86

mart_r · 2021-12-04, 10:47

A nice variation. The idea of e.g. consecutive gaps with certain merit value went through my head before, but I never got around to calculate specific numbers. I think that would fall out as a corollary of your table.
I would suggest continuing this search. You also see, for example, the gaps in twin primes depicted by long runs of the same value in the column x=4.

Dr Sardonicus · 2021-12-04, 14:38

I note that for a given x, the sequence of least primes with all g's $\ge$ x for increasing n (down a given column of the table) is monotone increasing; likewise for a given n and increasing lower bound x on gap size (along a given row).

The repeated values along the rows and columns are interesting - there are "jumps" in the gap size for given n, and in the number of consecutive primes with given minimum gap size.

For n = 1, the gap size has to be even to for (p, p + g₁) to be an "admissible" pair. For n+1 > 2 consecutive primes, the gap values are further constrained by the n + 1-tuple (p, p + g₁, p + g₁ + g₂, ...) being "admissible." In particular, if "x" is not divisible by 3, the triple (p, p + x, p + 2*x) is not admissible (one of them is divisible by 3 for p > 3).

I suppose one could look at n-tuples of gaps (g₁, g₂,... g_n) with all g's $\ge$ x, (but not much larger) for which (p, p + g₁, p + g₁ + g₂, ... p + g₁ + ... + g_n) is admissible.

One can probably use the Hardy-Littlewood conjectural asymptotic formula to estimate how large p has to be for there to be a decent chance of the tuple to consist entirely of primes; I don't know about the condition that they be consecutive primes.

Bobby Jacobs · 2021-12-05, 22:11

This is very cool!

robert44444uk · 2021-12-06, 10:16

I'm taking all gaps up to 50 in case anyone is interested in extending other ranges. 60 looks a soft target, as are all 0mod30 cases

What I find gratifying about this search is that the higher you go, you do find new records. The small gap cases are very easy to extend.

For example - 4

Code:

robert44444uk · 2021-12-13, 16:24

Quote:

Originally Posted by mart_r

A nice variation. The idea of e.g. consecutive gaps with certain merit value went through my head before, but I never got around to calculate specific numbers. I think that would fall out as a corollary of your table.

Just out of interest, I calculated the first few ranges of consecutive gaps, each with merit >1. The starting primes are not on OEIS

Code:

Min	merit:	1	Run	of	merits:	1	Initial	prime:	2	Following	gaps:	1	Merits:	1.44																												
Min	merit:	1	Run	of	merits:	2	Initial	prime:	2	Following	gaps:	1	2	Merits:	1.44	1.820478453																										
Min	merit:	1	Run	of	merits:	3	Initial	prime:	2	Following	gaps:	1	2	2	Merits:	1.44	1.820478453	1.242669869																								
Min	merit:	1	Run	of	merits:	4	Initial	prime:	2	Following	gaps:	1	2	2	4	Merits:	1.44	1.820478453	1.242669869	2.055593369																						
Min	merit:	1	Run	of	merits:	5	Initial	prime:	683	Following	gaps:	8	10	8	10	8	Merits:	1.225772819	1.529487021	1.220906581	1.523494836	1.216200763																				
Min	merit:	1	Run	of	merits:	6	Initial	prime:	1789	Following	gaps:	12	10	12	8	16	14	Merits:	1.602261949	1.3340275	1.599651392	1.065496216	2.129750371	1.861375897																		
Min	merit:	1	Run	of	merits:	7	Initial	prime:	13477	Following	gaps:	10	12	14	10	14	16	14	Merits:	1.051664069	1.261898449	1.472077185	1.05136911	1.471802284	1.681876797	1.471459487																
Min	merit:	1	Run	of	merits:	8	Initial	prime:	13477	Following	gaps:	10	12	14	10	14	16	14	10	Merits:	1.051664069	1.261898449	1.472077185	1.05136911	1.471802284	1.681876797	1.471459487	1.05092845														
Min	merit:	1	Run	of	merits:	9	Initial	prime:	13477	Following	gaps:	10	12	14	10	14	16	14	10	14	Merits:	1.051664069	1.261898449	1.472077185	1.05136911	1.471802284	1.681876797	1.471459487	1.05092845	1.47118591												
Min	merit:	1	Run	of	merits:	10	Initial	prime:	611561	Following	gaps:	26	16	18	20	16	14	22	14	22	24	Merits:	1.951399644	1.200857488	1.350962021	1.501065596	1.20084953	1.050741276	1.651162026	1.050736635	1.651154732	1.801254846										
Min	merit:	1	Run	of	merits:	11	Initial	prime:	611561	Following	gaps:	26	16	18	20	16	14	22	14	22	24	38	Merits:	1.951399644	1.200857488	1.350962021	1.501065596	1.20084953	1.050741276	1.651162026	1.050736635	1.651154732	1.801254846	2.851978442								
Min	merit:	1	Run	of	merits:	12	Initial	prime:	876853	Following	gaps:	18	22	20	16	18	24	32	24	16	14	16	18	Merits:	1.315395755	1.607703511	1.461545967	1.169234825	1.315387424	1.753847268	2.338458347	1.753839084	1.169223717	1.023069389	1.169220795	1.315371641						
Min	merit:	1	Run	of	merits:	13	Initial	prime:	876853	Following	gaps:	18	22	20	16	18	24	32	24	16	14	16	18	18	Merits:	1.315395755	1.607703511	1.461545967	1.169234825	1.315387424	1.753847268	2.338458347	1.753839084	1.169223717	1.023069389	1.169220795	1.315371641	1.315369668				
Min	merit:	1	Run	of	merits:	14	Initial	prime:	15022753	Following	gaps:	18	22	18	20	22	20	40	24	26	28	36	32	22	56	Merits:	1.089253658	1.33130993	1.089253482	1.210281559	1.331309608	1.210281355	2.420562514	1.452337275	1.573365229	1.694393146	2.178505227	1.93644881	1.331308385	3.388784681		
Min	merit:	1	Run	of	merits:	15	Initial	prime:	20917291	Following	gaps:	22	18	18	18	42	32	18	24	34	36	44	22	24	18	42	Merits:	1.305166519	1.067863449	1.067863395	1.06786334	2.491681	1.898423393	1.067863061	1.423817343	2.017074431	2.135725662	2.610331099	1.305165387	1.423816696	1.06786245	2.491678922

robert44444uk · 2021-12-14, 17:33

Quote:

Originally Posted by robert44444uk

Just out of interest, I calculated the first few ranges of consecutive gaps, each with merit >1. The starting primes are not on OEIS

Code:

Min	merit:	1	Run	of	merits:	1	Initial	prime:	2	Following	gaps:	1	Merits:	1.44																												
Min	merit:	1	Run	of	merits:	2	Initial	prime:	2	Following	gaps:	1	2	Merits:	1.44	1.820478453																										
Min	merit:	1	Run	of	merits:	3	Initial	prime:	2	Following	gaps:	1	2	2	Merits:	1.44	1.820478453	1.242669869																								
Min	merit:	1	Run	of	merits:	4	Initial	prime:	2	Following	gaps:	1	2	2	4	Merits:	1.44	1.820478453	1.242669869	2.055593369																						
Min	merit:	1	Run	of	merits:	5	Initial	prime:	683	Following	gaps:	8	10	8	10	8	Merits:	1.225772819	1.529487021	1.220906581	1.523494836	1.216200763																				
Min	merit:	1	Run	of	merits:	6	Initial	prime:	1789	Following	gaps:	12	10	12	8	16	14	Merits:	1.602261949	1.3340275	1.599651392	1.065496216	2.129750371	1.861375897																		
Min	merit:	1	Run	of	merits:	7	Initial	prime:	13477	Following	gaps:	10	12	14	10	14	16	14	Merits:	1.051664069	1.261898449	1.472077185	1.05136911	1.471802284	1.681876797	1.471459487																
Min	merit:	1	Run	of	merits:	8	Initial	prime:	13477	Following	gaps:	10	12	14	10	14	16	14	10	Merits:	1.051664069	1.261898449	1.472077185	1.05136911	1.471802284	1.681876797	1.471459487	1.05092845														
Min	merit:	1	Run	of	merits:	9	Initial	prime:	13477	Following	gaps:	10	12	14	10	14	16	14	10	14	Merits:	1.051664069	1.261898449	1.472077185	1.05136911	1.471802284	1.681876797	1.471459487	1.05092845	1.47118591												
Min	merit:	1	Run	of	merits:	10	Initial	prime:	611561	Following	gaps:	26	16	18	20	16	14	22	14	22	24	Merits:	1.951399644	1.200857488	1.350962021	1.501065596	1.20084953	1.050741276	1.651162026	1.050736635	1.651154732	1.801254846										
Min	merit:	1	Run	of	merits:	11	Initial	prime:	611561	Following	gaps:	26	16	18	20	16	14	22	14	22	24	38	Merits:	1.951399644	1.200857488	1.350962021	1.501065596	1.20084953	1.050741276	1.651162026	1.050736635	1.651154732	1.801254846	2.851978442								
Min	merit:	1	Run	of	merits:	12	Initial	prime:	876853	Following	gaps:	18	22	20	16	18	24	32	24	16	14	16	18	Merits:	1.315395755	1.607703511	1.461545967	1.169234825	1.315387424	1.753847268	2.338458347	1.753839084	1.169223717	1.023069389	1.169220795	1.315371641						
Min	merit:	1	Run	of	merits:	13	Initial	prime:	876853	Following	gaps:	18	22	20	16	18	24	32	24	16	14	16	18	18	Merits:	1.315395755	1.607703511	1.461545967	1.169234825	1.315387424	1.753847268	2.338458347	1.753839084	1.169223717	1.023069389	1.169220795	1.315371641	1.315369668				
Min	merit:	1	Run	of	merits:	14	Initial	prime:	15022753	Following	gaps:	18	22	18	20	22	20	40	24	26	28	36	32	22	56	Merits:	1.089253658	1.33130993	1.089253482	1.210281559	1.331309608	1.210281355	2.420562514	1.452337275	1.573365229	1.694393146	2.178505227	1.93644881	1.331308385	3.388784681		
Min	merit:	1	Run	of	merits:	15	Initial	prime:	20917291	Following	gaps:	22	18	18	18	42	32	18	24	34	36	44	22	24	18	42	Merits:	1.305166519	1.067863449	1.067863395	1.06786334	2.491681	1.898423393	1.067863061	1.423817343	2.017074431	2.135725662	2.610331099	1.305165387	1.423816696	1.06786245	2.491678922

Checked up to p=1001200549 and a second run of 15 found but no run of 16 yet.

Min merit: 1 Run of merits: 15 Initial prime: 34398523 Following gaps: 40 56 40 18 26 36 18 22 20 18 34 56 22 24 24 Merits: 2.30500730414805 3.22701000956899 2.3050069334542 1.03725305054945 1.49825436117103 2.07450594818822 1.03725291153983 1.2677535203211 1.15250315781684 1.03725280728285 1.95925524356651 3.22700845266168 1.26775320175876 1.38300344185791 1.3830033862545

Because the median merit following a prime is only about 0.7 merit - i.e. 0.7*g/ln(p) - there may be a case for saying that arbitrarily long runs of gaps with merit >1 do not occur. Interest to hear from others.

robert44444uk · 2021-12-16, 09:02

I found the first instance of two consecutive 300 gaps.

2 ||| 1362810282139 1362810282439 1362810282781 || 300 342

In the search I discovered 6 such. The largest gap following the 300 pair found was:

2 ||| 1724709287369 1724709287671 1724709287999 || 302 328 80

robert44444uk · 2021-12-20, 14:54

some findings about strings of minimum merits:

Merit >2

Code:

Min	merit:	2	Run	of	merits:	2	Initial	prime:	199	Following	gaps:	12	12							Merits:	2.267014728	2.242211901						
Min	merit:	2	Run	of	merits:	3	Initial	prime:	7351	Following	gaps:	18	24	18						Merits:	2.021883147	2.695103818	2.020590063					
Min	merit:	2	Run	of	merits:	4	Initial	prime:	648449	Following	gaps:	32	28	54	44					Merits:	2.391211347	2.092302213	4.03514125	3.287872413				
Min	merit:	2	Run	of	merits:	5	Initial	prime:	2352323	Following	gaps:	34	30	50	30	40				Merits:	2.317512986	2.044862385	3.408101012	2.044857645	2.726474489			
Min	merit:	2	Run	of	merits:	6	Initial	prime:	30097967	Following	gaps:	54	42	54	66	42	60			Merits:	3.135894622	2.439028896	3.135894041	3.832758984	2.439028134	3.484325623		
Min	merit:	2	Run	of	merits:	7	Initial	prime:	335739713	Following	gaps:	40	84	42	44	40	56	40		Merits:	2.03750573	4.278762006	2.139380976	2.241256246	2.037505665	2.852507913	2.037505635	
Min	merit:	2	Run	of	merits:	8	Initial	prime:	1065726007	Following	gaps:	42	48	52	54	48	50	76	50	Merits:	2.02050115	2.309144167	2.501572842	2.597787176	2.30914415	2.405358485	3.656144889	2.405358471

Merit >3

Code:

Min	merit:	3	Run	of	merits:	2	Initial	prime:	38461	Following	gaps:	40	42				Merits:	3.788811632	3.977860556			
Min	merit:	3	Run	of	merits:	3	Initial	prime:	740749	Following	gaps:	52	48	42			Merits:	3.847458022	3.551481267	3.107531211		
Min	merit:	3	Run	of	merits:	4	Initial	prime:	14784977	Following	gaps:	66	50	52	60		Merits:	3.997790871	3.028628629	3.149773129	3.634352836	
Min	merit:	3	Run	of	merits:	5	Initial	prime:	754473523	Following	gaps:	126	88	62	84	66	Merits:	6.163921998	4.304961361	3.033040941	4.109281259	3.228720971

Merit >4

Code:

Min	merit:	4	Run	of	merits:	2	Initial	prime:	413299	Following	gaps:	54	58			Merits:	4.175711922	4.484978606		
Min	merit:	4	Run	of	merits:	3	Initial	prime:	112219777	Following	gaps:	78	106	114		Merits:	4.208034635	5.718610957	6.150203923	
Min	merit:	4	Run	of	merits:	4	Initial	prime:	5289589577	Following	gaps:	180	92	90	100	Merits:	8.039660002	4.10915955	4.019829992	4.466477765

Min merit >5

Code:

Min	merit:	5	Run	of	merits:	2	Initial	prime:	10938023	Following	gaps:	102	96		Merits:	6.293287056	5.923090292	
Min	merit:	5	Run	of	merits:	3	Initial	prime:	1440754039	Following	gaps:	122	130	108	Merits:	5.785162115	6.164516983	5.12129101

2021-12-03, 12:06	#2
robert44444uk Jun 2003 Oxford, UK 2,039 Posts	The first instance of 7 consecutive 100+ gaps is 5183073661943 with gaps 116,130,114,158,286,102,102 The first instance of 3 consecutive 200+ gaps is provided by 20222645954633 with gaps of 204,216,218 Last fiddled with by robert44444uk on 2021-12-03 at 12:09

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2021-12-04, 08:10	#3
robert44444uk Jun 2003 Oxford, UK 2,039 Posts	The first instance of 13 and 14 consecutive 50+ gaps is at 3057601284499 where subsequent gaps are 60, 54, 58, 50, 100, 90, 62, 156, 124, 60, 60, 68, 76 and 86

2021-12-04, 10:47	#4
mart_r Dec 2008 you know...around... 2²×11×17 Posts	A nice variation. The idea of e.g. consecutive gaps with certain merit value went through my head before, but I never got around to calculate specific numbers. I think that would fall out as a corollary of your table. I would suggest continuing this search. You also see, for example, the gaps in twin primes depicted by long runs of the same value in the column x=4.

2021-12-04, 14:38	#5
Dr Sardonicus Feb 2017 Nowhere 3·1,931 Posts	I note that for a given x, the sequence of least primes with all g's $\ge$ x for increasing n (down a given column of the table) is monotone increasing; likewise for a given n and increasing lower bound x on gap size (along a given row). The repeated values along the rows and columns are interesting - there are "jumps" in the gap size for given n, and in the number of consecutive primes with given minimum gap size. For n = 1, the gap size has to be even to for (p, p + g₁) to be an "admissible" pair. For n+1 > 2 consecutive primes, the gap values are further constrained by the n + 1-tuple (p, p + g₁, p + g₁ + g₂, ...) being "admissible." In particular, if "x" is not divisible by 3, the triple (p, p + x, p + 2x) is not admissible (one of them is divisible by 3 for p > 3). I suppose one could look at n-tuples of gaps (g₁, g₂,... g_n) with all g's $\ge$ x, (but not much larger) for which (p, p + g₁, p + g₁ + g₂, ... p + g₁ + ... + g_n) is admissible. One can probably use the Hardy-Littlewood conjectural asymptotic formula to estimate how large p has to be for there to be a decent chance of the tuple to consist entirely of primes; I don't know about the condition that they be consecutive* primes.

2021-12-05, 22:11	#6
Bobby Jacobs May 2018 259₁₀ Posts	This is very cool!

2021-12-16, 09:02	#10
robert44444uk Jun 2003 Oxford, UK 2,039 Posts	I found the first instance of two consecutive 300 gaps. 2 \|\|\| 1362810282139 1362810282439 1362810282781 \|\| 300 342 In the search I discovered 6 such. The largest gap following the 300 pair found was: 2 \|\|\| 1724709287369 1724709287671 1724709287999 \|\| 302 328 80