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Old 2019-03-24, 21:34   #1
robert44444uk
 
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Default 2,4,6,8 what do we appreciate?

Anybody fancy extending these sequences?

https://www.primepuzzles.net/puzzles/puzz_011.htm
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Old 2019-03-25, 10:09   #2
Thomas11
 
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This sounds indeed interesting.

I played a bit with Perl and was able to generate the following distinct consecutive gaps within a few seconds:

21: 69931991 (2) 69931993 (10) 69932003 (8) 69932011 (30) 69932041 (18) 69932059 (22) 69932081 (32) 69932113 (16) 69932129 (42) 69932171 (36) 69932207 (20) 69932227 (46) 69932273 (6) 69932279 (44) 69932323 (48) 69932371 (40) 69932411 (26) 69932437 (12) 69932449 (54) 69932503 (4) 69932507 (50) 69932557

22: 203674907 (30) 203674937 (12) 203674949 (20) 203674969 (24) 203674993 (4) 203674997 (32) 203675029 (34) 203675063 (14) 203675077 (16) 203675093 (6) 203675099 (62) 203675161 (18) 203675179 (22) 203675201 (8) 203675209 (58) 203675267 (42) 203675309 (38) 203675347 (40) 203675387 (2) 203675389 (10) 203675399 (48) 203675447 (26) 203675473

23: 1092101119 (48) 1092101167 (24) 1092101191 (30) 1092101221 (58) 1092101279 (20) 1092101299 (4) 1092101303 (74) 1092101377 (52) 1092101429 (12) 1092101441 (2) 1092101443 (114) 1092101557 (22) 1092101579 (18) 1092101597 (42) 1092101639 (44) 1092101683 (10) 1092101693 (8) 1092101701 (36) 1092101737 (46) 1092101783 (56) 1092101839 (90) 1092101929 (28) 1092101957 (6) 1092101963
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Old 2019-03-26, 07:05   #3
robert44444uk
 
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Quote:
Originally Posted by Thomas11 View Post
This sounds indeed interesting.

I played a bit with Perl and was able to generate the following distinct consecutive gaps within a few seconds:

21: 69931991 (2) 69931993 (10) 69932003 (8) 69932011 (30) 69932041 (18) 69932059 (22) 69932081 (32) 69932113 (16) 69932129 (42) 69932171 (36) 69932207 (20) 69932227 (46) 69932273 (6) 69932279 (44) 69932323 (48) 69932371 (40) 69932411 (26) 69932437 (12) 69932449 (54) 69932503 (4) 69932507 (50) 69932557

22: 203674907 (30) 203674937 (12) 203674949 (20) 203674969 (24) 203674993 (4) 203674997 (32) 203675029 (34) 203675063 (14) 203675077 (16) 203675093 (6) 203675099 (62) 203675161 (18) 203675179 (22) 203675201 (8) 203675209 (58) 203675267 (42) 203675309 (38) 203675347 (40) 203675387 (2) 203675389 (10) 203675399 (48) 203675447 (26) 203675473

23: 1092101119 (48) 1092101167 (24) 1092101191 (30) 1092101221 (58) 1092101279 (20) 1092101299 (4) 1092101303 (74) 1092101377 (52) 1092101429 (12) 1092101441 (2) 1092101443 (114) 1092101557 (22) 1092101579 (18) 1092101597 (42) 1092101639 (44) 1092101683 (10) 1092101693 (8) 1092101701 (36) 1092101737 (46) 1092101783 (56) 1092101839 (90) 1092101929 (28) 1092101957 (6) 1092101963
It wont take long to get to Gusev's 36 record. Computers have come a long way since 2001!
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Old 2019-03-26, 08:40   #4
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Quote:
Originally Posted by robert44444uk View Post
It wont take long to get to Gusev's 36 record. Computers have come a long way since 2001!
Indeed, over night I was able to repeat Gusev's search:

24: 1363592621 (56) 1363592677 (10) 1363592687 (2) 1363592689 (84) 1363592773 (30) 1363592803 (58) 1363592861 (8) 1363592869 (60) 1363592929 (18) 1363592947 (12) 1363592959 (54) 1363593013 (24) 1363593037 (52) 1363593089 (42) 1363593131 (20) 1363593151 (96) 1363593247 (4) 1363593251 (26) 1363593277 (46) 1363593323 (14) 1363593337 (22) 1363593359 (32) 1363593391 (6) 1363593397 (40) 1363593437

25: 1363592677 (10) 1363592687 (2) 1363592689 (84) 1363592773 (30) 1363592803 (58) 1363592861 (8) 1363592869 (60) 1363592929 (18) 1363592947 (12) 1363592959 (54) 1363593013 (24) 1363593037 (52) 1363593089 (42) 1363593131 (20) 1363593151 (96) 1363593247 (4) 1363593251 (26) 1363593277 (46) 1363593323 (14) 1363593337 (22) 1363593359 (32) 1363593391 (6) 1363593397 (40) 1363593437 (56) 1363593493 (16) 1363593509

26: 2124140323 (48) 2124140371 (40) 2124140411 (152) 2124140563 (46) 2124140609 (2) 2124140611 (28) 2124140639 (14) 2124140653 (4) 2124140657 (72) 2124140729 (12) 2124140741 (20) 2124140761 (58) 2124140819 (140) 2124140959 (78) 2124141037 (30) 2124141067 (16) 2124141083 (56) 2124141139 (42) 2124141181 (10) 2124141191 (6) 2124141197 (62) 2124141259 (60) 2124141319 (34) 2124141353 (36) 2124141389 (24) 2124141413 (108) 2124141521

27: 23024158649 (8) 23024158657 (96) 23024158753 (60) 23024158813 (40) 23024158853 (54) 23024158907 (26) 23024158933 (66) 23024158999 (4) 23024159003 (18) 23024159021 (2) 23024159023 (10) 23024159033 (30) 23024159063 (24) 23024159087 (20) 23024159107 (22) 23024159129 (14) 23024159143 (88) 23024159231 (36) 23024159267 (62) 23024159329 (34) 23024159363 (38) 23024159401 (42) 23024159443 (64) 23024159507 (44) 23024159551 (12) 23024159563 (6) 23024159569 (48) 23024159617

28: 30282104173 (76) 30282104249 (8) 30282104257 (4) 30282104261 (38) 30282104299 (18) 30282104317 (36) 30282104353 (46) 30282104399 (30) 30282104429 (2) 30282104431 (22) 30282104453 (6) 30282104459 (32) 30282104491 (16) 30282104507 (74) 30282104581 (78) 30282104659 (10) 30282104669 (14) 30282104683 (34) 30282104717 (20) 30282104737 (52) 30282104789 (84) 30282104873 (26) 30282104899 (60) 30282104959 (64) 30282105023 (44) 30282105067 (12) 30282105079 (70) 30282105149 (50) 30282105199

29: 30282104173 (76) 30282104249 (8) 30282104257 (4) 30282104261 (38) 30282104299 (18) 30282104317 (36) 30282104353 (46) 30282104399 (30) 30282104429 (2) 30282104431 (22) 30282104453 (6) 30282104459 (32) 30282104491 (16) 30282104507 (74) 30282104581 (78) 30282104659 (10) 30282104669 (14) 30282104683 (34) 30282104717 (20) 30282104737 (52) 30282104789 (84) 30282104873 (26) 30282104899 (60) 30282104959 (64) 30282105023 (44) 30282105067 (12) 30282105079 (70) 30282105149 (50) 30282105199 (202) 30282105401

30: 196948778371 (16) 196948778387 (26) 196948778413 (4) 196948778417 (80) 196948778497 (34) 196948778531 (50) 196948778581 (18) 196948778599 (40) 196948778639 (162) 196948778801 (90) 196948778891 (20) 196948778911 (106) 196948779017 (12) 196948779029 (38) 196948779067 (22) 196948779089 (54) 196948779143 (44) 196948779187 (46) 196948779233 (48) 196948779281 (8) 196948779289 (30) 196948779319 (10) 196948779329 (2) 196948779331 (28) 196948779359 (78) 196948779437 (36) 196948779473 (24) 196948779497 (60) 196948779557 (66) 196948779623 (56) 196948779679

31: 263552821783 (76) 263552821859 (8) 263552821867 (34) 263552821901 (36) 263552821937 (26) 263552821963 (106) 263552822069 (38) 263552822107 (82) 263552822189 (18) 263552822207 (54) 263552822261 (50) 263552822311 (16) 263552822327 (62) 263552822389 (70) 263552822459 (14) 263552822473 (48) 263552822521 (42) 263552822563 (64) 263552822627 (6) 263552822633 (104) 263552822737 (24) 263552822761 (12) 263552822773 (66) 263552822839 (22) 263552822861 (30) 263552822891 (96) 263552822987 (2) 263552822989 (10) 263552822999 (74) 263552823073 (4) 263552823077 (32) 263552823109

32: 691340780027 (56) 691340780083 (66) 691340780149 (18) 691340780167 (16) 691340780183 (44) 691340780227 (42) 691340780269 (28) 691340780297 (102) 691340780399 (14) 691340780413 (90) 691340780503 (64) 691340780567 (84) 691340780651 (36) 691340780687 (32) 691340780719 (52) 691340780771 (2) 691340780773 (70) 691340780843 (20) 691340780863 (46) 691340780909 (50) 691340780959 (48) 691340781007 (4) 691340781011 (30) 691340781041 (60) 691340781101 (26) 691340781127 (132) 691340781259 (10) 691340781269 (12) 691340781281 (138) 691340781419 (8) 691340781427 (24) 691340781451 (6) 691340781457

33: 1625800359439 (22) 1625800359461 (50) 1625800359511 (102) 1625800359613 (40) 1625800359653 (26) 1625800359679 (82) 1625800359761 (96) 1625800359857 (72) 1625800359929 (32) 1625800359961 (112) 1625800360073 (36) 1625800360109 (2) 1625800360111 (30) 1625800360141 (48) 1625800360189 (70) 1625800360259 (8) 1625800360267 (24) 1625800360291 (16) 1625800360307 (62) 1625800360369 (34) 1625800360403 (6) 1625800360409 (12) 1625800360421 (20) 1625800360441 (52) 1625800360493 (14) 1625800360507 (60) 1625800360567 (42) 1625800360609 (88) 1625800360697 (74) 1625800360771 (10) 1625800360781 (38) 1625800360819 (4) 1625800360823 (56) 1625800360879

34: 1625800359439 (22) 1625800359461 (50) 1625800359511 (102) 1625800359613 (40) 1625800359653 (26) 1625800359679 (82) 1625800359761 (96) 1625800359857 (72) 1625800359929 (32) 1625800359961 (112) 1625800360073 (36) 1625800360109 (2) 1625800360111 (30) 1625800360141 (48) 1625800360189 (70) 1625800360259 (8) 1625800360267 (24) 1625800360291 (16) 1625800360307 (62) 1625800360369 (34) 1625800360403 (6) 1625800360409 (12) 1625800360421 (20) 1625800360441 (52) 1625800360493 (14) 1625800360507 (60) 1625800360567 (42) 1625800360609 (88) 1625800360697 (74) 1625800360771 (10) 1625800360781 (38) 1625800360819 (4) 1625800360823 (56) 1625800360879 (58) 1625800360937

35: 1625800359439 (22) 1625800359461 (50) 1625800359511 (102) 1625800359613 (40) 1625800359653 (26) 1625800359679 (82) 1625800359761 (96) 1625800359857 (72) 1625800359929 (32) 1625800359961 (112) 1625800360073 (36) 1625800360109 (2) 1625800360111 (30) 1625800360141 (48) 1625800360189 (70) 1625800360259 (8) 1625800360267 (24) 1625800360291 (16) 1625800360307 (62) 1625800360369 (34) 1625800360403 (6) 1625800360409 (12) 1625800360421 (20) 1625800360441 (52) 1625800360493 (14) 1625800360507 (60) 1625800360567 (42) 1625800360609 (88) 1625800360697 (74) 1625800360771 (10) 1625800360781 (38) 1625800360819 (4) 1625800360823 (56) 1625800360879 (58) 1625800360937 (134) 1625800361071

36: 1625800359439 (22) 1625800359461 (50) 1625800359511 (102) 1625800359613 (40) 1625800359653 (26) 1625800359679 (82) 1625800359761 (96) 1625800359857 (72) 1625800359929 (32) 1625800359961 (112) 1625800360073 (36) 1625800360109 (2) 1625800360111 (30) 1625800360141 (48) 1625800360189 (70) 1625800360259 (8) 1625800360267 (24) 1625800360291 (16) 1625800360307 (62) 1625800360369 (34) 1625800360403 (6) 1625800360409 (12) 1625800360421 (20) 1625800360441 (52) 1625800360493 (14) 1625800360507 (60) 1625800360567 (42) 1625800360609 (88) 1625800360697 (74) 1625800360771 (10) 1625800360781 (38) 1625800360819 (4) 1625800360823 (56) 1625800360879 (58) 1625800360937 (134) 1625800361071 (28) 1625800361099

Tested to 2,000,000,000,000.
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Old 2019-03-26, 18:12   #5
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Quote:
Originally Posted by Thomas11 View Post
Indeed, over night I was able to repeat Gusev's search:

...

33: 1625800359439 (22) ...
34: 1625800359439 (22) ...
35: 1625800359439 (22) ...

36: 1625800359439 (22) 1625800359461 (50) 1625800359511 (102) 1625800359613 (40) 1625800359653 (26) 1625800359679 (82) 1625800359761 (96) 1625800359857 (72) 1625800359929 (32) 1625800359961 (112) 1625800360073 (36) 1625800360109 (2) 1625800360111 (30) 1625800360141 (48) 1625800360189 (70) 1625800360259 (8) 1625800360267 (24) 1625800360291 (16) 1625800360307 (62) 1625800360369 (34) 1625800360403 (6) 1625800360409 (12) 1625800360421 (20) 1625800360441 (52) 1625800360493 (14) 1625800360507 (60) 1625800360567 (42) 1625800360609 (88) 1625800360697 (74) 1625800360771 (10) 1625800360781 (38) 1625800360819 (4) 1625800360823 (56) 1625800360879 (58) 1625800360937 (134) 1625800361071 (28) 1625800361099

Tested to 2,000,000,000,000.
With 16258... candidate performing admirably, the first 37 might take a little searching. I wonder what Gusev's maximum searched value was?
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Old 2019-04-03, 08:43   #6
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Meanwhile I've searched the distinct consecutive gaps up to 2*10^13 without finding anything better than N=36. Perhaps some of you wants to continue from there.

This is the Perl code I was using. Usage is:
Code:
perl consecutive_gaps.pl 2 1000000
Code:
#!/usr/bin/env perl
use warnings;
use strict;

use Math::Prime::Util qw/:all/;
use Time::HiRes qw(gettimeofday tv_interval);

my $start = shift || 2;
my $end = shift || 1000000;

my $starttime = [gettimeofday];

my $length = 0;

my @gaplist;
my @primelist;

my $prime = next_prime($start-1);

forprimes
{
  my $nextprime = $_;
  my $gap = $nextprime - $prime;
  my $index = -1;
  foreach my $i (0 .. $#gaplist)
  {
    if ($gap == $gaplist[$i])
    {
      $index = $i;
      last;
    }
  } 
  while ($index > -1)
  {
    shift @gaplist;
    shift @primelist;
    $index = $index - 1;
  }

  push @gaplist, $gap;
  push @primelist, $prime;

  if ($#gaplist > $length)
  {
    printf("%d: ", $#gaplist+1);
    foreach my $i (0 .. $#gaplist)
    {
      printf("%d (%d) ", $primelist[$i], $gaplist[$i]);
    }
    printf("%d\n", $nextprime);
    printf("\n");

    $length = $#gaplist;
  }

  $prime = $nextprime;
} $prime+1, $end;

my $duration = tv_interval($starttime);
printf("Execution time: %.2fs\n", $duration);
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Old 2019-04-03, 08:56   #7
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Thanks Thomas11. I'll take "consecutive different gaps" to 1e14

Last fiddled with by robert44444uk on 2019-04-03 at 10:00
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Old 2019-04-04, 08:17   #8
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Quote:
Originally Posted by robert44444uk View Post
Thanks Thomas11. I'll take "consecutive different gaps" to 1e14
Up to 2.2e13 and three sets of 34 but no 35, let alone a 37!

The Gusev 16258...sequence really is a thing of wonder.
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Old 2019-04-04, 08:36   #9
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Quote:
Originally Posted by robert44444uk View Post
Up to 2.2e13 and three sets of 34 but no 35, let alone a 37!

The Gusev 16258...sequence really is a thing of wonder.
Indeed!

I had only two sets of 35 between 1e13 and 2e13:

35: 12725917641061 (16) ...
35: 15779501807989 (24) ...
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Old 2019-04-04, 15:52   #10
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I wonder if someone can do the maths on this. I think one value for the chain of gaps would tend to be the most probable and it won't be length 1 or 2 chain. The most popular value will I am sure change and get larger, while the length champion will tend to increase at a fair rate of knots compared to ln(p).

I would hypothesise that there should be more than one jumping champion.

Just calculating the likelihood of the next prime being different is not easy. At the 2e14 to 2.2e14 level:

1 in 4 30 chains is also a 31
1 in 4 31 chains is also a 32

These are not statistically significant

1 in 3 32 chains is a 33
1 in 8 33 chains is a 34

The gap between 30 chains is 7.5e10 in this range , i.e. 1500 or so in the 2e13 between 2e14 and 2.2e14

Last fiddled with by robert44444uk on 2019-04-04 at 15:54
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Old 2019-04-05, 12:41   #11
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Quote:
Originally Posted by Thomas11 View Post
Indeed!

I had only two sets of 35 between 1e13 and 2e13:

35: 12725917641061 (16) ...
35: 15779501807989 (24) ...
Here are the next two:

35: 24536651667719 (50)…
35: 24965939258081 (66)...

Last fiddled with by robert44444uk on 2019-04-05 at 16:07
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