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 robert44444uk 2021-07-20 10:40

Repeating gaps

With the group's current computing resources it should be possible to take the following sequence and extend it

[url]https://oeis.org/A052187[/url]

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

 Bobby Jacobs 2021-07-22 13:26

[QUOTE=robert44444uk;583584]
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.
[/QUOTE]

There is also the repeating gap of 2 between the primes 3, 5, 7.

 Raydex 2021-07-24 07:45

[QUOTE=robert44444uk;583584]With the group's current computing resources it should be possible to take the following sequence and extend it

[url]https://oeis.org/A052187[/url]

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[/QUOTE]

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

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

[url]https://en.wikipedia.org/wiki/Balanced_prime[/url]

 robert44444uk 2021-07-26 09:37

[QUOTE=Raydex;583858]A minor correction to your original comment: Repeated gaps are all 0mod6 [B]except for the triplet 3, 5, 7.[/B]

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

[url]https://en.wikipedia.org/wiki/Balanced_prime[/url][/QUOTE]

Ah!! Of course! I forgot the law of small integers.

It is also a CPAP-3.

 robert44444uk 2021-07-27 14:23

I've completed 5e13 to 6e13, with no double gap >312.

I'm now taking 6e13 to 3e14.

 robert44444uk 2021-07-29 08:16

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!

 robert44444uk 2021-08-03 09:32

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[/CODE]

Smallest first instance gaps to find are now 366,378,390,396,402,408,414

 robert44444uk 2021-08-06 14:30

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.

 Bobby Jacobs 2021-08-06 19:51

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?

 rudy235 2021-08-07 08:19

[QUOTE=Bobby Jacobs;585007]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?[/QUOTE]

Probably not.:smile:

 robert44444uk 2021-08-08 14:07

[QUOTE=Bobby Jacobs;585007]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?[/QUOTE]

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

[/CODE]

 robert44444uk 2021-08-09 17:13

[QUOTE=robert44444uk;584684]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[/CODE]

Smallest first instance gaps to find are now 366,378,390,396,402,408,414[/QUOTE]

I'm up to 8.3e14 and taking this through to 1e15. 420 was repeated in the range just checked, but with no other 400's

New first gaps:

[CODE]
700926505323491 700926505323857 700926505324223 366
829317525163213 829317525163591 829317525163969 378
[/CODE]

 robert44444uk 2021-08-14 07:53

I completed the CPAP 3s up to 1e15, where I will stop. There were no further first instance examples to report. The largest examples in the range 87e13 to 100e13 were a pair of 378s.

I'm looking now at CPAP 4 and CPAP 5s, with a view to taking each to 1e15. At present I am at approx 2e13.

I have already found this unknown first instance beauty CPAP 5, all prime, all are separated by 120 all intervening integers are composite.

6683799874901 6683799875021 6683799875141 6683799875261 6683799875381

The first instance CPAP 5 30s and 60s were known, but not the first instance 90

112740413623 112740413713 112740413803 112740413893 112740413983

 robert44444uk 2021-08-16 11:53

Some CPAP 4 results to 1e14, first instance gaps (which are multiples of 6) I found found are as follows:

[CODE]
144 2988385487729 2988385487873 2988385488017 2988385488161
150 2665078848911 2665078849061 2665078849211 2665078849361
156 2867966168491 2867966168647 2867966168803 2867966168959
162 15559179029347 15559179029509 15559179029671 15559179029833
168 6213483902443 6213483902611 6213483902779 6213483902947
174 10435932748939 10435932749113 10435932749287 10435932749461
180 16006204702031 16006204702211 16006204702391 16006204702571
186 53461936795321 53461936795507 53461936795693 53461936795879
192 84217066456967 84217066457159 84217066457351 84217066457543
198 55463759865203 55463759865401 55463759865599 55463759865797
204 15178802276249 15178802276453 15178802276657 15178802276861
210
216
222 52418103824077 52418103824299 52418103824521 52418103824743

[/CODE]

This extends [URL="http://http://oeis.org/A052239"]http://oeis.org/A052239[/URL]

it is notable there are no 210 gap CPAP 4s in the range from 2 to 1e14 - a requirement for CPAPs larger than CPAP 6 is that all gaps must be a multiple of 210.

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