mersenneforum.org Repeating gaps
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 2021-07-20, 10:40 #1 robert44444uk     Jun 2003 Oxford, UK 19·103 Posts Repeating gaps With the group's current computing resources it should be possible to take the following sequence and extend it https://oeis.org/A052187 So 47,53,59 is the first sequence of primes with a repeated gap of 6. Repeated gaps are all 0mod6. This is easy to prove. The highest known members of the sequence are 46186474937633 46186474937957 46186474938281, differing by 324. However no repeating gap of 318 is know below 5e13 according to Johnson and Resta. I might have a look at 5e13 to 6e13
2021-07-22, 13:26   #2
Bobby Jacobs

May 2018

111010012 Posts

Quote:
 Originally Posted by robert44444uk So 47,53,59 is the first sequence of primes with a repeated gap of 6. Repeated gaps are all 0mod6. This is easy to prove.
There is also the repeating gap of 2 between the primes 3, 5, 7.

2021-07-24, 07:45   #3
Raydex

Nov 2020
Massachusetts, USA

23×3 Posts

Quote:
 Originally Posted by robert44444uk With the group's current computing resources it should be possible to take the following sequence and extend it https://oeis.org/A052187 So 47,53,59 is the first sequence of primes with a repeated gap of 6. Repeated gaps are all 0mod6. This is easy to prove. The highest known members of the sequence are 46186474937633 46186474937957 46186474938281, differing by 324. However no repeating gap of 318 is know below 5e13 according to Johnson and Resta. I might have a look at 5e13 to 6e13

A minor correction to your original comment: Repeated gaps are all 0mod6 except for the triplet 3, 5, 7.

Also, I believe the name for this is "balanced primes." You can read more about that here.

https://en.wikipedia.org/wiki/Balanced_prime

2021-07-26, 09:37   #4
robert44444uk

Jun 2003
Oxford, UK

19·103 Posts

Quote:
 Originally Posted by Raydex A minor correction to your original comment: Repeated gaps are all 0mod6 except for the triplet 3, 5, 7. Also, I believe the name for this is "balanced primes." You can read more about that here. https://en.wikipedia.org/wiki/Balanced_prime
Ah!! Of course! I forgot the law of small integers.

It is also a CPAP-3.

 2021-07-27, 14:23 #5 robert44444uk     Jun 2003 Oxford, UK 7A516 Posts I've completed 5e13 to 6e13, with no double gap >312. I'm now taking 6e13 to 3e14.
 2021-07-29, 08:16 #6 robert44444uk     Jun 2003 Oxford, UK 19·103 Posts I'm up to 1.284e14 and I have found the following first instance double gaps 79280544407833 79280544408151 79280544408469 318 65110291176851 65110291177181 65110291177511 330 66798158654761 66798158655097 66798158655433 336 75949420736251 75949420736599 75949420736947 348 74613428569219 74613428569591 74613428569963 372 91546947852407 91546947852797 91546947853187 390 So no first instance 342, 354, surprisingly no 360, 366, 378, and 384. On, on to the first >400!
 2021-08-03, 09:32 #7 robert44444uk     Jun 2003 Oxford, UK 19·103 Posts Tested now up to 4.1e14 and a first gap of 420 has appeared! New first instance double gaps: Code: 203340719517647 203340719517989 203340719518331 342 227795281925129 227795281925483 227795281925837 354 130520768860387 130520768860747 130520768861107 360 269099493634513 269099493634897 269099493635281 384 187891466722493 187891466722913 187891466723333 420 Smallest first instance gaps to find are now 366,378,390,396,402,408,414
 2021-08-06, 14:30 #8 robert44444uk     Jun 2003 Oxford, UK 19·103 Posts 4.1e14 to 5.5e14 checked, with no further first instance double gaps. In fact the largest in this range was only 360. I'm doing 5.5e14 to 6.9e14 now.
 2021-08-06, 19:51 #9 Bobby Jacobs     May 2018 233 Posts I find it interesting that the first prime gap of 12 is immediately followed by another gap of 12. The first gap of 12 is between 199 and 211, and the next prime after 211 is 223. Are there any numbers n other than 2 and 12 such that the first prime gap of n is followed by another gap of n?
2021-08-07, 08:19   #10
rudy235

Jun 2015
Vallejo, CA/.

2×5×101 Posts

Quote:
 Originally Posted by Bobby Jacobs I find it interesting that the first prime gap of 12 is immediately followed by another gap of 12. The first gap of 12 is between 199 and 211, and the next prime after 211 is 223. Are there any numbers n other than 2 and 12 such that the first prime gap of n is followed by another gap of n?
Probably not.

2021-08-08, 14:07   #11
robert44444uk

Jun 2003
Oxford, UK

19·103 Posts

Quote:
 Originally Posted by Bobby Jacobs I find it interesting that the first prime gap of 12 is immediately followed by another gap of 12. The first gap of 12 is between 199 and 211, and the next prime after 211 is 223. Are there any numbers n other than 2 and 12 such that the first prime gap of n is followed by another gap of n?
Here is a table of first prime gaps and first repeating prime gaps. I would agree with rudy235 on this, although it is not impossible that another example exists, just exceedingly unlikely. But look at the gap of 144 - "only" 32 times the size of the first gap of that size.

A Gap size, B first repeating, C first instance gap of that size, D ratio of B to C

Code:
A       B       C      D
6	47	23	2.0
12	199	199	1.0
18	20183	523	38.6
24	16763	1669	10.0
30	69593	4297	16.2
36	255767	9551	26.8
42	247099	16141	15.3
48	3565931	28229	126.3
54	6314393	35617	177.3
60	4911251	43331	113.3
66	12012677	162143	74.1
72	23346737	31397	743.6
78	43607351	188029	231.9
84	34346203	461717	74.4
90	36598517	404851	90.4
96	51041957	360653	141.5
102	460475467	1444309	318.8
108	652576321	2238823	291.5
114	742585183	492113	1509.0
120	530324329	1895359	279.8
126	807620651	1671781	483.1
132	2988119207	1357201	2201.7
138	12447231761	3826019	3253.3
144	383204539	11981443	32.0
150	4470607951	13626257	328.1
156	5007182707	17983717	278.4
162	36589015439	39175217	934.0
168	5558570323	37305713	149.0
174	88526967673	52721113	1679.2
180	65997364441	17051707	3870.4
186	48287689531	147684137	327.0
192	57484162139	123454691	465.6
198	50284155091	46006769	1093.0
204	178796541613	112098817	1595.0
210	264860525297	20831323	12714.5
216	978720895037	202551667	4832.0
222	472446412199	122164747	3867.3
228	374787490691	895858039	418.4
234	1550817881693	189695659	8175.3
240	1521870803867	391995431	3882.4
246	2930505040471	555142061	5278.8
252	3427394302919	630045137	5439.9
258	2093308790593	1316355323	1590.2
264	4842748395259	2357881993	2053.9
270	4934972897089	1391048047	3547.7
276	4228611064261	649580171	6509.8
282	10639324457617	436273009	24386.9
288	19968161343593	1294268491	15428.1
294	22281535886963	5692630189	3914.1
300	6537587646371	4758958741	1373.7
306	17432065861211	3917587237	4449.7
312	28219476363139	6570018347	4295.2

Last fiddled with by robert44444uk on 2021-08-09 at 15:47

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