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Old 2019-03-03, 09:12   #1
robert44444uk
 
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Default Gap symmetry, n-interprimes, primorials

I wonder if anyone has done any systemic work on gap symmetry? By this I mean there are consecutive gaps that are symmetrical in length around a central point. This is equivalent to saying there are primes symmetrically distant from a central point.

Pioneer work has been done - and the central point has been dubbed "interprime"

The simplest is defined half way between any two primes - each prime is equidistant from the central point. This series of numbers is defines as (p(1)+p(2))/2, and this is listed on OEIS as interprimes - https://oeis.org/A024675

For two primes either side of the central point the series is shown as https://oeis.org/A263674 double interprimes 9,12,15,18,30,42... where, for example primes 5 and 7 are 4 and 2 distant from 9, as are 13 and 11

For three primes either side, the series is 12,15,30,42,105,144... where, for example primes 41,37 and 31 and 1,4, and 11 away from 42, as are 53,47 and 43 . This series is not listed on Sloane

The list for 4-interprimes looks quite odd! 15645,19425,34485,34845,35988,46641...

5-interprimes..783630 is the only one up I could find up to pi(100000). Consecutive nearby primes are 783569 (61 away), 783571 (59 away), 783599 (31 away), 783613 (17 away), 783619 (11 away), 783641 (11 away), 783647 (17 away), 783661 (31 away), 783677 (59 away) and 783689 (61 away). The gap sequence is 2,28,14,6,22,6,14,28,2

Dirichlet I'm sure gives us the insight that there are infinitely many n-interprimes for any value of n.

Current computing power would suggest that it should be possible to find 10-interprimes (i.e. a list of 20 primes)

I also noted that although there are 70 or so lists of interprimes on OEIS, none look at primorials and multiples of primorials. I would have thought these would have provided a good grounding for finding n-interprimes, given their modular symmetry around the centre point. "close to" primorials i.e. p#/x, where x is a squarefree p-smooth number are also good hunting ground.

As examples:
the 3-interprime series listed above 12,15,30,42,105,144,165,312 could be restated as:
2*3#... 5#/2...5#...7#/5...7#/2...24*3#...11#/14... 52*3#.

However, near primorials are less impressive for the 4-interprime list:

15645,19425,34485,34845,35988,46641 could be restated as:

149*7#/2...185*7#/2...209*11#/14..2323*5#/2...5998*3#...2221*7#/10

Last fiddled with by robert44444uk on 2019-03-03 at 09:14
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Old 2019-03-04, 08:39   #2
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Quote:
Originally Posted by robert44444uk View Post

The list for 4-interprimes looks quite odd! 15645,19425,34485,34845,35988,46641...

5-interprimes..783630 is the only one up I could find up to pi(100000). Consecutive nearby primes are 783569 (61 away), 783571 (59 away), 783599 (31 away), 783613 (17 away), 783619 (11 away), 783641 (11 away), 783647 (17 away), 783661 (31 away), 783677 (59 away) and 783689 (61 away). The gap sequence is 2,28,14,6,22,6,14,28,2
I should have programmed this rather than rely on Microsoft Excel formulae:

Here are the lists for 4-, 5-, 6-, and 7-interprimes. Col 1 shows the interprime, and the following columns are the relevant primes:

Code:
4-interprimes to 1e5
30 17 19 23 29 31 37 41 43
165 149 151 157 163 167 173 179 181
705 677 683 691 701 709 719 727 733
870 853 857 859 863 877 881 883 887
1290 1277 1279 1283 1289 1291 1297 1301 1303
5460 5437 5441 5443 5449 5471 5477 5479 5483
6132 6101 6113 6121 6131 6133 6143 6151 6163
13545 13499 13513 13523 13537 13553 13567 13577 13591
13965 13921 13931 13933 13963 13967 13997 13999 14009
19890 19853 19861 19867 19889 19891 19913 19919 19927
22101 22073 22079 22091 22093 22109 22111 22123 22129
41895 41863 41879 41887 41893 41897 41903 41911 41927
49704 49667 49669 49681 49697 49711 49727 49739 49741
51345 51307 51329 51341 51343 51347 51349 51361 51383
51828 51797 51803 51817 51827 51829 51839 51853 51859
55818 55799 55807 55813 55817 55819 55823 55829 55837
61662 61637 61643 61651 61657 61667 61673 61681 61687
66360 66337 66343 66347 66359 66361 66373 66377 66383
83250 83227 83231 83233 83243 83257 83267 83269 83273
91140 91121 91127 91129 91139 91141 91151 91153 91159

5-interprimes to 1e7:
30 13 17 19 23 29 31 37 41 43 47
165 139 149 151 157 163 167 173 179 181 191
6132 6091 6101 6113 6121 6131 6133 6143 6151 6163 6173
19890 19843 19853 19861 19867 19889 19891 19913 19919 19927 19937
51828 51787 51797 51803 51817 51827 51829 51839 51853 51859 51869
55818 55793 55799 55807 55813 55817 55819 55823 55829 55837 55843
113160 113143 113147 113149 113153 113159 113161 113167 113171 113173 113177
179070 179029 179033 179041 179051 179057 179083 179089 179099 179107 179111
205212 205157 205171 205187 205201 205211 205213 205223 205237 205253 205267
302580 302551 302563 302567 302573 302579 302581 302587 302593 302597 302609
346395 346361 346369 346373 346391 346393 346397 346399 346417 346421 346429
460980 460949 460951 460969 460973 460979 460981 460987 460991 461009 461011
895851 895799 895801 895813 895823 895841 895861 895879 895889 895901 895903
970515 970447 970457 970469 970481 970493 970537 970549 970561 970573 970583
1150695 1150651 1150657 1150661 1150673 1150687 1150703 1150717 1150729 1150733 1150739
1180875 1180847 1180849 1180853 1180859 1180873 1180877 1180891 1180897 1180901 1180903
1697304 1697257 1697261 1697287 1697291 1697299 1697309 1697317 1697321 1697347 1697351
1929585 1929553 1929559 1929563 1929569 1929581 1929589 1929601 1929607 1929611 1929617
2334852 2334781 2334793 2334803 2334823 2334841 2334863 2334881 2334901 2334911 2334923
2580660 2580631 2580647 2580649 2580653 2580659 2580661 2580667 2580671 2580673 2580689
2797485 2797447 2797453 2797463 2797477 2797481 2797489 2797493 2797507 2797517 2797523
3056625 3056561 3056579 3056593 3056597 3056611 3056639 3056653 3056657 3056671 3056689
3086055 3086009 3086011 3086021 3086033 3086047 3086063 3086077 3086089 3086099 3086101
3416757 3416717 3416731 3416741 3416747 3416753 3416761 3416767 3416773 3416783 3416797
3598995 3598943 3598949 3598957 3598967 3598981 3599009 3599023 3599033 3599041 3599047
4024695 4024667 4024673 4024679 4024687 4024693 4024697 4024703 4024711 4024717 4024723
4026165 4026107 4026109 4026131 4026137 4026149 4026181 4026193 4026199 4026221 4026223
4067175 4067123 4067137 4067143 4067149 4067171 4067179 4067201 4067207 4067213 4067227
4077630 4077583 4077607 4077611 4077617 4077629 4077631 4077643 4077649 4077653 4077677
4389540 4389503 4389509 4389521 4389523 4389533 4389547 4389557 4389559 4389571 4389577
4541118 4541083 4541087 4541099 4541107 4541113 4541123 4541129 4541137 4541149 4541153
4790565 4790503 4790521 4790537 4790551 4790557 4790573 4790579 4790593 4790609 4790627
5050644 5050597 5050601 5050607 5050609 5050631 5050657 5050679 5050681 5050687 5050691
5191380 5191339 5191349 5191363 5191367 5191369 5191391 5191393 5191397 5191411 5191421
5202690 5202641 5202643 5202647 5202653 5202671 5202709 5202727 5202733 5202737 5202739
5327343 5327303 5327317 5327323 5327327 5327339 5327347 5327359 5327363 5327369 5327383
5658525 5658473 5658479 5658491 5658493 5658503 5658547 5658557 5658559 5658571 5658577
5674662 5674601 5674607 5674621 5674631 5674661 5674663 5674693 5674703 5674717 5674723
5687850 5687813 5687827 5687833 5687837 5687839 5687861 5687863 5687867 5687873 5687887
5741015 5740949 5740967 5740979 5740991 5740997 5741033 5741039 5741051 5741063 5741081
5771295 5771263 5771267 5771281 5771287 5771291 5771299 5771303 5771309 5771323 5771327
6419031 6418991 6419003 6419009 6419011 6419029 6419033 6419051 6419053 6419059 6419071
6437403 6437317 6437329 6437363 6437383 6437393 6437413 6437423 6437443 6437477 6437489
6554241 6554153 6554173 6554201 6554213 6554239 6554243 6554269 6554281 6554309 6554329
6580785 6580727 6580729 6580733 6580753 6580757 6580813 6580817 6580837 6580841 6580843
7129239 7129211 7129217 7129219 7129229 7129231 7129247 7129249 7129259 7129261 7129267
7275972 7275923 7275937 7275943 7275967 7275971 7275973 7275977 7276001 7276007 7276021
7484337 7484297 7484303 7484311 7484317 7484333 7484341 7484357 7484363 7484371 7484377
7987509 7987457 7987459 7987477 7987481 7987487 7987531 7987537 7987541 7987559 7987561
8021811 8021759 8021771 8021789 8021791 8021801 8021821 8021831 8021833 8021851 8021863
8548176 8548121 8548129 8548151 8548159 8548171 8548181 8548193 8548201 8548223 8548231
8569335 8569283 8569289 8569313 8569321 8569331 8569339 8569349 8569357 8569381 8569387
8750907 8750867 8750873 8750881 8750891 8750893 8750921 8750923 8750933 8750941 8750947
9195585 9195539 9195541 9195551 9195559 9195583 9195587 9195611 9195619 9195629 9195631
9515655 9515593 9515621 9515641 9515647 9515651 9515659 9515663 9515669 9515689 9515717

6-interprimes to 1e8:
165 137 139 149 151 157 163 167 173 179 181 191 193
55818 55787 55793 55799 55807 55813 55817 55819 55823 55829 55837 55843 55849
113160 113131 113143 113147 113149 113153 113159 113161 113167 113171 113173 113177 113189
179070 179021 179029 179033 179041 179051 179057 179083 179089 179099 179107 179111 179119
895851 895789 895799 895801 895813 895823 895841 895861 895879 895889 895901 895903 895913
1150695 1150649 1150651 1150657 1150661 1150673 1150687 1150703 1150717 1150729 1150733 1150739 1150741
3086055 3086003 3086009 3086011 3086021 3086033 3086047 3086063 3086077 3086089 3086099 3086101 3086107
4026165 4026103 4026107 4026109 4026131 4026137 4026149 4026181 4026193 4026199 4026221 4026223 4026227
4077630 4077559 4077583 4077607 4077611 4077617 4077629 4077631 4077643 4077649 4077653 4077677 4077701
8021811 8021753 8021759 8021771 8021789 8021791 8021801 8021821 8021831 8021833 8021851 8021863 8021869
8750907 8750857 8750867 8750873 8750881 8750891 8750893 8750921 8750923 8750933 8750941 8750947 8750957
12577110 12577063 12577079 12577087 12577091 12577099 12577109 12577111 12577121 12577129 12577133 12577141 12577157
14355600 14355559 14355563 14355569 14355571 14355577 14355581 14355619 14355623 14355629 14355631 14355637 14355641
19136589 19136527 19136531 19136539 19136561 19136569 19136581 19136597 19136609 19136617 19136639 19136647 19136651
19412937 19412863 19412867 19412873 19412881 19412917 19412933 19412941 19412957 19412993 19413001 19413007 19413011
20066025 20065961 20065963 20065987 20066003 20066009 20066023 20066027 20066041 20066047 20066063 20066087 20066089
21865389 21865339 21865351 21865357 21865367 21865379 21865381 21865397 21865399 21865411 21865421 21865427 21865439
22633182 22633141 22633147 22633151 22633153 22633157 22633181 22633183 22633207 22633211 22633213 22633217 22633223
25880220 25880177 25880189 25880191 25880203 25880207 25880213 25880227 25880233 25880237 25880249 25880251 25880263
30405039 30404971 30404977 30405007 30405019 30405029 30405031 30405047 30405049 30405059 30405071 30405101 30405107
33926256 33926159 33926161 33926171 33926209 33926231 33926251 33926261 33926281 33926303 33926341 33926351 33926353
38202255 38202173 38202191 38202209 38202221 38202233 38202253 38202257 38202277 38202289 38202301 38202319 38202337
41905950 41905891 41905907 41905909 41905921 41905933 41905939 41905961 41905967 41905979 41905991 41905993 41906009
42925785 42925699 42925717 42925721 42925733 42925781 42925783 42925787 42925789 42925837 42925849 42925853 42925871
43746177 43746077 43746097 43746121 43746137 43746163 43746167 43746187 43746191 43746217 43746233 43746257 43746277
44635605 44635537 44635543 44635561 44635583 44635589 44635603 44635607 44635621 44635627 44635649 44635667 44635673
55057485 55057417 55057433 55057439 55057463 55057477 55057481 55057489 55057493 55057507 55057531 55057537 55057553
61738350 61738273 61738291 61738297 61738301 61738319 61738349 61738351 61738381 61738399 61738403 61738409 61738427
62550600 62550563 62550571 62550577 62550581 62550583 62550589 62550611 62550617 62550619 62550623 62550629 62550637
63200130 63200069 63200083 63200089 63200117 63200119 63200129 63200131 63200141 63200143 63200171 63200177 63200191
76011780 76011731 76011751 76011763 76011769 76011773 76011779 76011781 76011787 76011791 76011797 76011809 76011829
79669965 79669897 79669903 79669921 79669943 79669949 79669951 79669979 79669981 79669987 79670009 79670027 79670033
91091610 91091563 91091573 91091579 91091591 91091599 91091603 91091617 91091621 91091629 91091641 91091647 91091657
91591500 91591441 91591447 91591463 91591469 91591477 91591481 91591519 91591523 91591531 91591537 91591553 91591559

7-interprimes to 1e8:
8021811 8021749 8021753 8021759 8021771 8021789 8021791 8021801 8021821 8021831 8021833 8021851 8021863 8021869 8021873
20066025 20065943 20065961 20065963 20065987 20066003 20066009 20066023 20066027 20066041 20066047 20066063 20066087 20066089 20066107
62550600 62550557 62550563 62550571 62550577 62550581 62550583 62550589 62550611 62550617 62550619 62550623 62550629 62550637 62550643

Another aspect - symmetrical gaps could be around a prime, whereas interprimes are always a composite. I'll stick to interprimes for now.

Last fiddled with by robert44444uk on 2019-03-04 at 08:54
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Old 2019-03-04, 09:41   #3
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The first 8-interprime: 1071065190

the first 9-interprime: 1613902650

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Old 2019-03-06, 09:41   #4
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I'm still looking for the first 10-interprime

Larger (for example 6-) interprimes predominantly appear to be divisible by 3.

In the first 896 6-interprimes, 2 are 1mod3 (3565765570 and 9032233630), 4 are 2mod3 (10273573520, 156990740, 8636267360, 9257290415) and the other 890 are 0mod3.

Not really understanding why

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Old 2019-03-06, 17:04   #5
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Quote:
Originally Posted by robert44444uk View Post
In the first 896 6-interprimes, 2 are 1mod3 (3565765570 and 9032233630), 4 are 2mod3 (10273573520, 156990740, 8636267360, 9257290415) and the other 890 are 0mod3.

Not really understanding why
One observation: If N is an interprime, N is not divisible by 3, and the smallest prime in the interval is greater than 3, then all the gaps must have length divisible by 3.

For if N =/= 0 (mod 3), p is one of the primes, p =/= N (mod 3), and p =/= 0 (mod 3), then the symmetrically located number 2*N - p will be == 0 (mod 3).

This condition might narrow things down a bit.
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Old 2019-03-07, 09:34   #6
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Quote:
Originally Posted by Dr Sardonicus View Post
One observation: If N is an interprime, N is not divisible by 3, and the smallest prime in the interval is greater than 3, then all the gaps must have length divisible by 3.

For if N =/= 0 (mod 3), p is one of the primes, p =/= N (mod 3), and p =/= 0 (mod 3), then the symmetrically located number 2*N - p will be == 0 (mod 3).

This condition might narrow things down a bit.
Neat! For 3565765770 with prime factors 2,5,356576577 the gaps are 6,6,24,6,18,18,18,6,24,6,6, all 0mod3.

It is easy to demonstrate in general that the majority of gaps are not 0mod3 in length - for example in the first 100000 gaps, the sum of 1mod3 and 2mod3 gaps is about 38.4% more than 0mod3 - and hence the longer (larger n) the n-interprime, the greater the requirement for all prime gaps involved in the interprime to be 0mod3, and subsequently the less likely the interprime is 1 or 2mod3. Using the 41.9% 0mod3 gap density in the first 100000 gaps as a given, the chances of 11 gaps in a row 0mod3 is about 14157:1 - this would provide for less 1mod3 and 2mod3 6-interprimes than are actually turning up 6 out of 896 is 149:1 - this could of course just be chance - but I wonder if I am again missing something here.

Does the ratio of the count of successive prime gaps (1mod3+2mod3)/0mod3 tend towards a constant? Can't see why not. Where can I look for work done on that?

I've tested up to 1.5e10 and, in addition to the 9-interprime mentioned above, there are two others:

74422046685
81661695390

There are 33 8-interprimes in the range 0 to 1.5e10

But no 10-interprime yet

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Old 2019-03-07, 14:08   #7
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If 3|N, and there are 2*k consecutive primes occurring symmetrically around N, k on each side, there can be up to k - 1 or whatever, of gaps with non-zero residues (mod 3) (between consecutive primes, except the first gap of p - N or N - p) on either side. But they have to occur in a specific sequence. The non-zero residues of gaps can be dispersed among any number of gaps of length divisible by 3, but on one side have to occur in the sequence

1,1,2,1,2,1,2,1,... (the 2,1 repeats)

and the nonzero residues (mod 3) of the corresponding gaps on the other side of N must occur in the sequence

2,2,1,2,1,2,1,2,... (the 1,2 repeats)

The number of possibilities obviously increases without bound as k increases without bound.

I'm too lazy to work out how fast. Either that, or I'm just not very good at this sort of thing
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Old 2019-03-07, 18:14   #8
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Obviously, the gaps, being symmetrical around N, will be the same on both sides of N as you move away from N. Equally obviously, if 3|N, the first gaps p-N and N-p are not divisible by 3. That'll teach me to post before my blood caffeine is up to working levels. At least until the next time I forget and do it again.

But the observation of two possible sequences of non-zero residues is right. And now, with sufficient caffeine in my system, I can try to figure out how many sequences of mod 3 residues are possible in a sequence of k gaps on one side of N, assuming 3|N. The first gap has to have a non-zero residue (mod 3). Say it's 1. Each of the other k-1 gaps can have either a zero residue, or a non-zero residue. The succeeding non-zero residues have to be from the repeating sequence 1,2,1,2,... so, having picked j of the remaining k-1 slots, which can be done in binomial (k-1,j) ways, there is only one way to fill them in with non-zero residues. Thus, having chosen the residue 1 for the first gap, there are 2^(k-1) ways to assign a sequence of k residues (mod 3). If we choose the initial residue 2 instead, we get another 2^(k-1) possibilities. So, 2^k in all.

Now, we look at the 2*k - 1 gaps between a sequence of 2*k consecutive primes. We'll assume we start with a prime p > 3. (In the case of 2*k primes symmetrically placed about N, the two gaps between N and the nearest primes are joined into one gap between consecutive primes.) There are 2 possible residues (mod 3) for the first gap. Choosing one, the remaining gaps (mod 3) can be filled in either with 0's, or the initial terms of whichever sequence of nonzero residues 1,2,1,2... or 2,1,2,1... avoids multiples of 3. By the same argument as before, the result is a total of 2^(2*k-1) possible sequences of residues (mod 3).

So, if 3|N, the symmetry condition of k primes either side of N restricts the sequences of residues (mod 3) to 2^k out of 2^(2*k-1) possibilities for a sequence of 2*k consecutive primes. If N is not divisible by 3 (and the smallest prime is greater than 3), the symmetry condition restricts consideration to one sequence (all zeroes) out of 2^(2*k-1) possible sequences of residues (mod 3) of the sequence of 2*k - 1 gaps between 2*k consecutive primes.

I have no clue whether the possible sequences of residues (mod 3) of of gaps between a given number of consecutive primes are in any sense "equally distributed."
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Old 2019-03-07, 21:37   #9
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I will have to read Dr S's post in the morning - wine is no help.

I'm a bit worried that I can't find literature on prime gaps mod3. Google isn't what it used to be. There appears to be bits 'n pieces of lit around gaps mod6, given that these are noticeably spiky on graphs of gap frequency over a given range.

A couple of statements I have read seem to suggest that the numbers of 0mod3 gaps is equal to 50% of gaps - but the evidence over the first 100000 gaps suggested 41.9% - this is surely far too large a difference for 50% to be the long run average.
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Old 2019-03-08, 03:06   #10
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Quote:
Originally Posted by robert44444uk View Post
I've tested up to 1.5e10 and, in addition to the 9-interprime mentioned above, there are two others:

74422046685
81661695390

There are 33 8-interprimes in the range 0 to 1.5e10

But no 10-interprime yet
1.5e10 or 1.5e11 (aka 150e9)?
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Old 2019-03-08, 08:49   #11
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Quote:
Originally Posted by axn View Post
1.5e10 or 1.5e11 (aka 150e9)?
Gosh, I am getting muddled with my zero's. It was 1.5e11 Now I am up to 5e11, and have three cores working on ranges up to 4e12.

Still no 10-interprime though.
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