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RE: [PrimeNumbers] Isomorphic Numbers

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  • Jon Perry
    7 : {684,685,686,687,688,689,690} 8 : {1081,1082,1083,1084,1085,1086,1087,1088} 9 : {1872,...1880} 11: {2419,...2429} 12: {9811,...9822} 13: {30650,...30662}
    Message 1 of 26 , Mar 4, 2003
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      '7 : {684,685,686,687,688,689,690}
      8 : {1081,1082,1083,1084,1085,1086,1087,1088}
      9 : {1872,...1880}
      11: {2419,...2429}
      12: {9811,...9822}
      13: {30650,...30662}
      14: {64027,...64040}
      15: {64029,...64043}
      16: {102035,...102050}
      17: {146467,...146483}
      18: {372879,...372896}
      22: {613787,...613808}

      22 elements is 11*11 odd-yielding pairs. So there's 121 coincidences in the
      above set. That can't be right, surely? '

      I suppose the 'coincidence level' depends on the proportion of primes in
      each range.

      Jon Perry
      perry@...
      http://www.users.globalnet.co.uk/~perry/maths/
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    • Phil Carmody
      ... Yeah, but even if the sums are expected 92.5% composite, and the products expected 96.25% composite, then randomly you d expect a single coincidence with
      Message 2 of 26 , Mar 4, 2003
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        --- Jon Perry <perry@...> wrote:
        > '7 : {684,685,686,687,688,689,690}
        > 8 : {1081,1082,1083,1084,1085,1086,1087,1088}
        > 9 : {1872,...1880}
        > 11: {2419,...2429}
        > 12: {9811,...9822}
        > 13: {30650,...30662}
        > 14: {64027,...64040}
        > 15: {64029,...64043}
        > 16: {102035,...102050}
        > 17: {146467,...146483}
        > 18: {372879,...372896}
        > 22: {613787,...613808}
        >
        > 22 elements is 11*11 odd-yielding pairs. So there's 121 coincidences in the
        > above set. That can't be right, surely? '
        >
        > I suppose the 'coincidence level' depends on the proportion of primes in
        > each range.

        Yeah, but even if the sums are expected 92.5% composite, and the products
        expected 96.25% composite, then randomly you'd expect a single coincidence
        with probability 0.893 and therefore 121 of them with probability 1/870200.

        OK, it looks more possible than I first thought, but still, the 23-element
        set seems too elusive (requiring 132 coincidences). I ran it for 20 times
        longer than the 22 above. I can only assume that I searched 20* further, but
        I didn't log how far I got. I was using GP/Pari, so a C/Pascal
        implementation should race through the possibilities much faster.

        It's surely someone else's turn to find one. Jack? Jens? Jud? Jon?
        Other people beginning with a J?

        Phil



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      • Jack Brennen
        Phil, I think there is a problem with your sets of numbers... you don t seem to have considered the case where a & b are both even, thus requiring that ab+1 be
        Message 3 of 26 , Mar 4, 2003
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          Phil, I think there is a problem with your sets of numbers... you don't
          seem to have considered the case where a & b are both even, thus
          requiring that ab+1 be an odd composite.

          For instance, you gave the range (684..690), but we have:

          684+688 is composite
          684*688+1 is prime

          But some of the ranges you gave are correct, by coincidence I guess. :)

          I get the following results up to length 20, which took some 2 minutes
          of CPU time using C with the GMP library... Note that someone might
          want to doublecheck these, since my primality testing wasn't rigorous,
          only probable:

          run length 2 -> (1..2)
          run length 3 -> (1..3)
          run length 4 -> (1..4)
          run length 5 -> (1..5)
          run length 6 -> (4..9)
          run length 7 -> (906..912)
          run length 8 -> (1081..1088)
          run length 9 -> (2212..2220)
          run length 10 -> (2419..2428)
          run length 11 -> (2419..2429)
          run length 12 -> (20153..20164)
          run length 13 -> (30650..30662)
          run length 14 -> (64027..64040)
          run length 15 -> (64029..64043)
          run length 16 -> (102035..102050)
          run length 17 -> (372879..372895)
          run length 18 -> (1060846..1060863)
          run length 19 -> (4895741..4895759)
          run length 20 -> (5385547..5385566)

          There is no run of length 21 below 10^7 (again, subject to the
          caveat of only using a probable prime test).

          Jack
        • jbrennen <jack@brennen.net>
          ... run length 21 - (15776479..15776499) No run of length 22 below 2.8*10^7. I ll let it run for a day or so before stopping the search. It s currently
          Message 4 of 26 , Mar 4, 2003
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            --- In primenumbers@yahoogroups.com, Jack Brennen <jack@b...> wrote:
            >
            > There is no run of length 21 below 10^7 (again, subject to the
            > caveat of only using a probable prime test).
            >

            run length 21 -> (15776479..15776499)

            No run of length 22 below 2.8*10^7.

            I'll let it run for a day or so before stopping the search.
            It's currently extending the search limit by about 14000/second.
          • Paul Jobling
            ... In which case I will stop mine, which is going at 8k/sec on this PC. But it agrees with your results so far.
            Message 5 of 26 , Mar 4, 2003
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              > run length 21 -> (15776479..15776499)
              >
              > No run of length 22 below 2.8*10^7.
              >
              > I'll let it run for a day or so before stopping the search.
              > It's currently extending the search limit by about 14000/second.

              In which case I will stop mine, which is going at 8k/sec on this PC. But it
              agrees with your results so far.


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            • Jon Perry
              Note that someone might want to doublecheck these, since my primality testing wasn t rigorous, only probable: They are all valid: { test(n,k)=
              Message 6 of 26 , Mar 4, 2003
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                'Note that someone might
                want to doublecheck these, since my primality testing wasn't rigorous,
                only probable:'

                They are all valid:

                {
                test(n,k)=
                v=vector(k,i,i+n-1);
                vl=length(v);fl=0;
                for (i=1,vl-1,for (j=i+1,vl,
                if (isprime(v[i]+v[j])!=isprime(v[i]*v[j]+1),fl=1;break);
                if (fl==1,break)));
                if (fl==0,print("ok"),print("fail"));
                }

                test(906,7)
                test(1081,8)
                test(2212,9)
                test(2419,10)
                test(2419,11)
                test(20153,12)
                test(30650,13)
                test(64027,14)
                test(64029,15)
                test(102035,16)
                test(372879,17)
                test(1060846,18)
                test(4895741,19)
                test(5385547,20)
                test(15776479,21)

                Jon Perry
                perry@...
                http://www.users.globalnet.co.uk/~perry/maths/
                http://www.users.globalnet.co.uk/~perry/DIVMenu/
                BrainBench MVP for HTML and JavaScript
                http://www.brainbench.com
              • Jon Perry
                Using: { test(n,k)= v=vector(k,i,i+n-1); vl=length(v);fl=0;pc=0; for (i=1,vl-1,for (j=i+1,vl, if (isprime(v[i]+v[j])!=isprime(v[i]*v[j]+1),fl=1;break); if
                Message 7 of 26 , Mar 4, 2003
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                  Using:

                  {
                  test(n,k)=
                  v=vector(k,i,i+n-1);
                  vl=length(v);fl=0;pc=0;
                  for (i=1,vl-1,for (j=i+1,vl,
                  if (isprime(v[i]+v[j])!=isprime(v[i]*v[j]+1),fl=1;break);
                  if (fl==1,break);
                  if (isprime(v[i]+v[j]),pc++)));
                  [fl,pc];
                  }

                  to check the current list yields 0 primes for all run length>7.

                  So I think we should adopt a policy of a least 1 prime from a range.

                  Jon Perry
                  perry@...
                  http://www.users.globalnet.co.uk/~perry/maths/
                  http://www.users.globalnet.co.uk/~perry/DIVMenu/
                  BrainBench MVP for HTML and JavaScript
                  http://www.brainbench.com
                • Jon Perry
                  The first with rl 8 is [5255:2] [9312:1] is the only other to 10000. Jon Perry perry@globalnet.co.uk http://www.users.globalnet.co.uk/~perry/maths/
                  Message 8 of 26 , Mar 4, 2003
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                    The first with rl 8 is

                    [5255:2]

                    [9312:1] is the only other to 10000.

                    Jon Perry
                    perry@...
                    http://www.users.globalnet.co.uk/~perry/maths/
                    http://www.users.globalnet.co.uk/~perry/DIVMenu/
                    BrainBench MVP for HTML and JavaScript
                    http://www.brainbench.com

                    -
                  • Phil Carmody
                    ... It correctly gave me the length 5 and 6 results, so I applied the strong law of small numbers, which I double-checked using engineers induction. That s
                    Message 9 of 26 , Mar 4, 2003
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                      --- Jack Brennen <jack@...> wrote:
                      > Phil, I think there is a problem with your sets of numbers... you don't
                      > seem to have considered the case where a & b are both even, thus
                      > requiring that ab+1 be an odd composite.

                      It correctly gave me the length 5 and 6 results, so I applied the strong law
                      of small numbers, which I double-checked using engineers' induction.

                      That's what caveats are for, I guess.

                      I shut down my emacs/shell/GP window, so I have no way of working out where
                      I went wrong, it was all ephemeral.

                      Phil


                      =====
                      "Only an admission that he does possess weapons of mass destruction
                      would do, sources said: 'The rest is just gesture politics." -- Hoon

                      "Are you still bombing your wife?" -- Winjer

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                    • jbrennen <jack@brennen.net>
                      ... In that case: run length 8 - (5255..5262) run length 9 - (8009..8017) run length 10 - (16268..16277) run length 11 - (16268..16278) run length 12 -
                      Message 10 of 26 , Mar 4, 2003
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                        --- In primenumbers@yahoogroups.com, "Jon Perry" <perry@g...> wrote:
                        >
                        > So I think we should adopt a policy of a least 1 prime from a range.
                        >

                        In that case:

                        run length 8 -> (5255..5262)
                        run length 9 -> (8009..8017)
                        run length 10 -> (16268..16277)
                        run length 11 -> (16268..16278)
                        run length 12 -> (118151..118162)
                        run length 13 -> (252035..252047)
                        run length 14 -> (267485..267498)
                        run length 15 -> (267485..267499)
                        run length 16 -> (305923..305938)

                        With no length 17 run less than 5*10^6.
                      • Jon Perry
                        Another challenge is to find to max. number of primes. rl8; ? for (n=10000,50000,x=test(n,8);if (!x[1] && x[2] 0,print1(n : x[2] , ))) 12106:2, 12775:2,
                        Message 11 of 26 , Mar 4, 2003
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                          Another challenge is to find to max. number of primes.

                          rl8;
                          ? for (n=10000,50000,x=test(n,8);if (!x[1] && x[2]>0,print1(n":"x[2]", ")))
                          12106:2, 12775:2, 13069:2, 15869:3, 15870:3, 16064:1, 16189:2, 16268:1,
                          17409:1,
                          17920:2, 17924:2, 18344:5, 18411:2, 18416:1, 18552:1, 19466:1, 21715:2,
                          21806:1
                          , 22820:1, 24689:1, 25956:1, 26113:2, 26865:1, 27304:3, 28479:3, 28480:2,
                          28481:
                          1, 29709:1, 29931:1, 31614:1, 31725:1, 34035:1, 34410:1, 34805:1, 35091:1,
                          35359
                          :2, 36204:1, 36205:2, 36633:2, 36634:3, 36716:1, 36784:2, 37520:1, 38829:1,
                          3924
                          7:2, 39296:1, 39759:1, 39760:2, 39935:2, 40403:3, 40404:3, 40748:1, 42866:1,
                          434
                          05:2, 43829:1, 45192:1, 46362:1, 47678:1, 48019:3, 48315:1, 49484:1,
                          49755:1, 49
                          789:2, 49790:1, 49953:1,

                          So 18344:5 is the current record.

                          Jon Perry
                          perry@...
                          http://www.users.globalnet.co.uk/~perry/maths/
                          http://www.users.globalnet.co.uk/~perry/DIVMenu/
                          BrainBench MVP for HTML and JavaScript
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                        • Phil Carmody
                          ... 2420+2420 = 4840 = composite 2420*2420+1=5856401 = prime Are the two elements supposed to be distinct? Jon doesn t state that in his frist post today. Phil
                          Message 12 of 26 , Mar 4, 2003
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                            --- Jack Brennen <jack@...> wrote:
                            > run length 10 -> (2419..2428)

                            2420+2420 = 4840 = composite
                            2420*2420+1=5856401 = prime

                            Are the two elements supposed to be distinct? Jon doesn't state
                            that in his frist post today.

                            Phil


                            =====
                            "Only an admission that he does possess weapons of mass destruction
                            would do, sources said: 'The rest is just gesture politics." -- Hoon

                            "Are you still bombing your wife?" -- Winjer

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                          • Phil Carmody
                            ... I agree with you on all previous data points, and this one too:
                            Message 13 of 26 , Mar 4, 2003
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                              --- "jbrennen <jack@...>" <jack@...> wrote:
                              > --- In primenumbers@yahoogroups.com, Jack Brennen <jack@b...> wrote:
                              > >
                              > > There is no run of length 21 below 10^7 (again, subject to the
                              > > caveat of only using a probable prime test).
                              > >
                              >
                              > run length 21 -> (15776479..15776499)

                              I agree with you on all previous data points, and this one too:
                              <<<
                              Starting program: /mnt/net/kilospaz/home/phil/projects/maths/primes/conjectures/perry/sumprod 1 21
                              [...]
                              t=60.450000 b=15728652)

                              Breakpoint 4, main (argc=3, argv=0x0) at sumprod.c:108
                              (gdb) next
                              (gdb)
                              (gdb)
                              15776479..15776499 is len 21
                              >>>

                              > No run of length 22 below 2.8*10^7.
                              >
                              > I'll let it run for a day or so before stopping the search.
                              > It's currently extending the search limit by about 14000/second.

                              The "t=60.45" means the program took 60.45 seconds of 900MHz Duron.
                              I'm currently clocking 200000/s, but expect to slow down as time goes on.

                              <<<
                              t=89.730000 b=37748755)

                              Breakpoint 4, main (argc=3, argv=0x0) at sumprod.c:108
                              (gdb) next
                              (gdb)
                              (gdb)
                              38392415..38392436 is len 22
                              >>>

                              90s, but that run didn't start from 0, but from the len=21 one.

                              So I'm looking at the 23 now, hopefully I'll be able to finish my sentence
                              before it find one...

                              ..., bleh.

                              Phil


                              =====
                              "Only an admission that he does possess weapons of mass destruction
                              would do, sources said: 'The rest is just gesture politics." -- Hoon

                              "Are you still bombing your wife?" -- Winjer

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                            • Phil Carmody
                              ... unbleh!
                              Message 14 of 26 , Mar 4, 2003
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                                --- Phil Carmody <thefatphil@...> wrote:
                                > <<<
                                > t=89.730000 b=37748755)
                                >
                                > Breakpoint 4, main (argc=3, argv=0x0) at sumprod.c:108
                                > (gdb) next
                                > (gdb)
                                > (gdb)
                                > 38392415..38392436 is len 22
                                > >>>
                                >
                                > 90s, but that run didn't start from 0, but from the len=21 one.
                                >
                                > So I'm looking at the 23 now, hopefully I'll be able to finish my sentence
                                > before it find one...
                                >
                                > ..., bleh.

                                unbleh!

                                <<<
                                t=156.960000 b=75497473)

                                Breakpoint 4, main (argc=3, argv=0x0) at sumprod.c:108
                                (gdb) next
                                (gdb)
                                (gdb)
                                76350646..76350668 is len 23
                                >>>

                                and instantly thereafter:
                                76350646..76350669 is len 24

                                Phil


                                =====
                                "Only an admission that he does possess weapons of mass destruction
                                would do, sources said: 'The rest is just gesture politics." -- Hoon

                                "Are you still bombing your wife?" -- Winjer

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                              • Phil Carmody
                                ... 370902159..370902183 is len 25 (1262 seconds, FWIW) I ll not look at the must include a prime version until I ve got to 2^31 and my signed ints run out
                                Message 15 of 26 , Mar 4, 2003
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                                  > 38392415..38392436 is len 22
                                  > 76350646..76350668 is len 23
                                  > 76350646..76350669 is len 24

                                  370902159..370902183 is len 25

                                  (1262 seconds, FWIW)

                                  I'll not look at the 'must include a prime' version until I've got to 2^31
                                  and my signed ints run out of juice.

                                  Phil


                                  =====
                                  "Only an admission that he does possess weapons of mass destruction
                                  would do, sources said: 'The rest is just gesture politics." -- Hoon

                                  "Are you still bombing your wife?" -- Winjer

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                                • Phil Carmody
                                  ... 3330 further seconds without the prime requirement provided: 1091876084..1091876109 is len 26 ... Agreed. 5901351..5901367 is len 17 (16s) 6120922..6120939
                                  Message 16 of 26 , Mar 5, 2003
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                                    --- "jbrennen <jack@...>" <jack@...> wrote:

                                    3330 further seconds without the prime requirement provided:
                                    1091876084..1091876109 is len 26

                                    > --- In primenumbers@yahoogroups.com, "Jon Perry" <perry@g...> wrote:
                                    > >
                                    > > So I think we should adopt a policy of a least 1 prime from a range.
                                    > >
                                    >
                                    > In that case:
                                    >
                                    > run length 8 -> (5255..5262)
                                    > run length 9 -> (8009..8017)
                                    > run length 10 -> (16268..16277)
                                    > run length 11 -> (16268..16278)
                                    > run length 12 -> (118151..118162)
                                    > run length 13 -> (252035..252047)
                                    > run length 14 -> (267485..267498)
                                    > run length 15 -> (267485..267499)
                                    > run length 16 -> (305923..305938)
                                    >
                                    > With no length 17 run less than 5*10^6.

                                    Agreed.

                                    5901351..5901367 is len 17 (16s)
                                    6120922..6120939 is len 18 (<1s)
                                    15736249..15736267 is len 19 (35s)
                                    15736249..15736268 is len 20 (instant)
                                    75952831..75952851 is len 21 (3m49s)
                                    311606294..311606315 is len 22 (15m9s)

                                    I'm just going to break the code^W^W^Wdo a few tweaks, and then I'll re-run
                                    them all, and push it further. 250000/s just isn't fast enough... ;-)

                                    Phil


                                    =====
                                    "Only an admission that he does possess weapons of mass destruction
                                    would do, sources said: 'The rest is just gesture politics." -- Hoon

                                    "Are you still bombing your wife?" -- Winjer

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                                  • Jon Perry
                                    Are the two elements supposed to be distinct? Jon doesn t state that in his first post today. This wasn t a condition in the original discussion, indeed we
                                    Message 17 of 26 , Mar 5, 2003
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                                      'Are the two elements supposed to be distinct? Jon doesn't state
                                      that in his first post today.'

                                      This wasn't a condition in the original discussion, indeed we have 4+4=8,
                                      4*4+1=17, same for 2.

                                      For stringency this could be added, although graphs with loops are frowned
                                      at...

                                      Jon Perry
                                      perry@...
                                      http://www.users.globalnet.co.uk/~perry/maths/
                                      http://www.users.globalnet.co.uk/~perry/DIVMenu/
                                      BrainBench MVP for HTML and JavaScript
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                                    • Phil Carmody
                                      ... You seem back-to-front, Jon. 8 is composite, 17 is prime and {1,2,3,4,5} was provided as a solution. Therefore the condition that the pair s elements are
                                      Message 18 of 26 , Mar 5, 2003
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                                        --- Jon Perry <perry@...> wrote:
                                        > 'Are the two elements supposed to be distinct? Jon doesn't state
                                        > that in his first post today.'
                                        >
                                        > This wasn't a condition in the original discussion, indeed we have 4+4=8,
                                        > 4*4+1=17, same for 2.

                                        You seem back-to-front, Jon.

                                        8 is composite, 17 is prime and {1,2,3,4,5} was provided as a solution.
                                        Therefore the condition that the pair's elements are distinct _was_ a
                                        condition in the first one?
                                        If it wasn't a condition, then {1,2,3,4,5} wouldn't have been a solution.

                                        > For stringency this could be added, although graphs with loops are frowned
                                        > at...

                                        Nope, trees and forests with loops are frowned upon (and any other DAGs, of
                                        course)

                                        Phil


                                        =====
                                        "Only an admission that he does possess weapons of mass destruction
                                        would do, sources said: 'The rest is just gesture politics." -- Hoon

                                        "Are you still bombing your wife?" -- Winjer

                                        __________________________________________________
                                        Do you Yahoo!?
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                                      • Phil Carmody
                                        ... 311606294..311606316 is len 23 (instant, doh!) 467282388..467282411 is len 24 (9m50s) 1142221381..1142221405 is len 25 (42m33s) It was only 4% faster
                                        Message 19 of 26 , Mar 5, 2003
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                                          --- Phil Carmody <thefatphil@...> wrote:
                                          > > run length 8 -> (5255..5262)
                                          > > run length 9 -> (8009..8017)
                                          > > run length 10 -> (16268..16277)
                                          > > run length 11 -> (16268..16278)
                                          > > run length 12 -> (118151..118162)
                                          > > run length 13 -> (252035..252047)
                                          > > run length 14 -> (267485..267498)
                                          > > run length 15 -> (267485..267499)
                                          > > run length 16 -> (305923..305938)
                                          > 5901351..5901367 is len 17 (16s)
                                          > 6120922..6120939 is len 18 (<1s)
                                          > 15736249..15736267 is len 19 (35s)
                                          > 15736249..15736268 is len 20 (instant)
                                          > 75952831..75952851 is len 21 (3m49s)
                                          > 311606294..311606315 is len 22 (15m9s)

                                          311606294..311606316 is len 23 (instant, doh!)
                                          467282388..467282411 is len 24 (9m50s)
                                          1142221381..1142221405 is len 25 (42m33s)

                                          It was only 4% faster looking for ones with primes in the range. So I think
                                          that for me it would make sense to look for all such runs, regardless of the
                                          existance of primes in the sum's range.

                                          Phil


                                          =====
                                          "Only an admission that he does possess weapons of mass destruction
                                          would do, sources said: 'The rest is just gesture politics." -- Hoon

                                          "Are you still bombing your wife?" -- Winjer

                                          __________________________________________________
                                          Do you Yahoo!?
                                          Yahoo! Tax Center - forms, calculators, tips, more
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                                        • Jon Perry
                                          These are for the isomorph a+b ab-1. (no guarenteed prime) run length 1 [1,...,1] run length 2 [2,...,3] run length 3 [5,...,7] run length 4 [56,...,59]
                                          Message 20 of 26 , Mar 5, 2003
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                                            These are for the isomorph a+b <=> ab-1.

                                            (no guarenteed prime)

                                            run length 1 [1,...,1]
                                            run length 2 [2,...,3]
                                            run length 3 [5,...,7]
                                            run length 4 [56,...,59]
                                            run length 5 [121,...,125]
                                            run length 6 [211,...,216]
                                            run length 7 [819,...,825]
                                            run length 8 [1470,...,1477]
                                            run length 9 [2231,...,2239]
                                            run length 10 [15886,...,15895]
                                            run length 11 [44275,...,44285]
                                            run length 12 [44275,...,44286]
                                            run length 13 [44275,...,44287]

                                            I left the code running for about an hour after I found rl13, so rl14 is
                                            high.

                                            Jon Perry
                                            perry@...
                                            http://www.users.globalnet.co.uk/~perry/maths/
                                            http://www.users.globalnet.co.uk/~perry/DIVMenu/
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                                          • Phil Carmody
                                            ... First ones with primes appended, and appended. ... 787374..787387 is len 14 1262441..1262455 is len 15 3623705..3623720 is len 16 7993838..7993854 is len
                                            Message 21 of 26 , Mar 5, 2003
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                                              --- Jon Perry <perry@...> wrote:
                                              > These are for the isomorph a+b <=> ab-1.
                                              >
                                              > (no guarenteed prime)

                                              First ones with primes appended, and appended.

                                              > run length 1 [1,...,1]
                                              > run length 2 [2,...,3]
                                              > run length 3 [5,...,7]
                                              > run length 4 [56,...,59]
                                              > run length 5 [121,...,125] 131..135
                                              > run length 6 [211,...,216]
                                              > run length 7 [819,...,825] 915..921
                                              > run length 8 [1470,...,1477] 1470..1477
                                              > run length 9 [2231,...,2239] 2231..2239
                                              > run length 10 [15886,...,15895] 42672..42681
                                              > run length 11 [44275,...,44285] 89663..89673
                                              > run length 12 [44275,...,44286] 89663..89674
                                              > run length 13 [44275,...,44287] 505989..506001

                                              787374..787387 is len 14
                                              1262441..1262455 is len 15
                                              3623705..3623720 is len 16
                                              7993838..7993854 is len 17
                                              7993838..7993855 is len 18
                                              7993838..7993856 is len 19
                                              7993838..7993857 is len 20
                                              117532072..117532092 is len 21 (6m40s)


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                                            • thefatphil
                                              ... 535716000..535716021 is len 22 (27m35s) 535716000..535716022 is len 23 I stopped it at ~1.4G after nearly an hour looking for a 24. Phil
                                              Message 22 of 26 , Mar 5, 2003
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                                                --- In primenumbers@yahoogroups.com, Phil Carmody <thefatphil@y...> wrote:
                                                > --- Jon Perry <perry@g...> wrote:
                                                > > These are for the isomorph a+b <=> ab-1.
                                                > >
                                                > > (no guarenteed prime)
                                                >
                                                > First ones with primes appended, and appended.
                                                >
                                                > > run length 1 [1,...,1]
                                                > > run length 2 [2,...,3]
                                                > > run length 3 [5,...,7]
                                                > > run length 4 [56,...,59]
                                                > > run length 5 [121,...,125] 131..135
                                                > > run length 6 [211,...,216]
                                                > > run length 7 [819,...,825] 915..921
                                                > > run length 8 [1470,...,1477] 1470..1477
                                                > > run length 9 [2231,...,2239] 2231..2239
                                                > > run length 10 [15886,...,15895] 42672..42681
                                                > > run length 11 [44275,...,44285] 89663..89673
                                                > > run length 12 [44275,...,44286] 89663..89674
                                                > > run length 13 [44275,...,44287] 505989..506001
                                                >
                                                > 787374..787387 is len 14
                                                > 1262441..1262455 is len 15
                                                > 3623705..3623720 is len 16
                                                > 7993838..7993854 is len 17
                                                > 7993838..7993855 is len 18
                                                > 7993838..7993856 is len 19
                                                > 7993838..7993857 is len 20
                                                > 117532072..117532092 is len 21 (6m40s)

                                                535716000..535716021 is len 22 (27m35s)
                                                535716000..535716022 is len 23

                                                I stopped it at ~1.4G after nearly an hour looking for a 24.

                                                Phil
                                              • Jon Perry
                                                Looking at: Check out his idea here: http://mnemo.nu/math/problems/?action=problem_view&id=107 I can t see how polytope has proved that all ab+1 are
                                                Message 23 of 26 , Mar 31, 2003
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                                                  Looking at:

                                                  'Check out his idea here:

                                                  http://mnemo.nu/math/problems/?action=problem_view&id=107'

                                                  I can't see how polytope has proved that all ab+1 are composites.

                                                  Indeed using this code:

                                                  {
                                                  forstep (n=4,20,2,v=vector(n/2);
                                                  for (i=1,n/2,v[i]=n!/2+i);
                                                  for (i=1,n/2,for (j=i+1,n/2,
                                                  if (isprime(v[i]*v[j]+1),print(n":"i","j)))))
                                                  }

                                                  yields the following exceptions.

                                                  16:4,7
                                                  18:3,6
                                                  18:5,8
                                                  20:7,10
                                                  20:8,9

                                                  (note 16:4,7 implies that (16!/2+4) * (16!/2+7) + 1 =
                                                  109440784174463838480384029 is prime)

                                                  We did adopt a policy of at least one prime in the thread, as the
                                                  all-composite sets seemed to be too easy; although we did not manage a proof
                                                  of the existence of arbitary sets as attempted by polytope.

                                                  http://groups.yahoo.com/group/primenumbers/message/11660

                                                  The sets mentions are not the first composite-only runs:

                                                  http://groups.yahoo.com/group/primenumbers/message/11670

                                                  And

                                                  http://groups.yahoo.com/group/primenumbers/message/11673

                                                  sums up to current knowledge of the problem.

                                                  Jon Perry
                                                  perry@...
                                                  http://www.users.globalnet.co.uk/~perry/maths/
                                                  http://www.users.globalnet.co.uk/~perry/DIVMenu/
                                                  BrainBench MVP for HTML and JavaScript
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