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Isomorphic Numbers

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  • Jon Perry
    Prompted by a thread Jack and myself had a while ago, you will recall that the set: S={1,2,3,4,5} has the property that a+b is prime a.b+1 is prime Can
    Message 1 of 26 , Mar 4, 2003
      Prompted by a thread Jack and myself had a while ago, you will recall that
      the set:

      S={1,2,3,4,5}

      has the property that a+b is prime <=> a.b+1 is prime

      Can anyone find a larger set of continuous (or not) integers with this
      property?

      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
      ... Not-continuous, I present { 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, .... You get the idea, I trust.
      Message 2 of 26 , Mar 4, 2003
        --- Jon Perry <perry@...> wrote:
        > Prompted by a thread Jack and myself had a while ago, you will recall that
        > the set:
        >
        > S={1,2,3,4,5}
        >
        > has the property that a+b is prime <=> a.b+1 is prime
        >
        > Can anyone find a larger set of continuous (or not) integers with this
        > property?

        Not-continuous, I present
        { 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39,
        41, 43, 45, 47, 49, ....

        You get the idea, I trust.

        Phil


        =====
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        would do, sources said: 'The rest is just gesture politics." -- Hoon

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      • Phil Carmody
        ... {4,5,6,7,8,9} seems to do the business for contiguous ranges. Spoilt, as ever, by the Carmody-prime 91. These I didn t use just pen and paper for, and
        Message 3 of 26 , Mar 4, 2003
          --- Jon Perry <perry@...> wrote:
          > Prompted by a thread Jack and myself had a while ago, you will recall that
          > the set:
          >
          > S={1,2,3,4,5}
          >
          > has the property that a+b is prime <=> a.b+1 is prime
          >
          > Can anyone find a larger set of continuous (or not) integers with this
          > property?

          {4,5,6,7,8,9} seems to do the business for contiguous ranges.

          Spoilt, as ever, by the Carmody-prime 91.

          These I didn't use just pen and paper for, and therefore are much more
          likely to be wrong -

          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?

          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
          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 4 of 26 , Mar 4, 2003
            '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 5 of 26 , Mar 4, 2003
              --- 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



              =====
              "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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            • 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 6 of 26 , Mar 4, 2003
                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 7 of 26 , Mar 4, 2003
                  --- 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 8 of 26 , Mar 4, 2003
                    > 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 9 of 26 , Mar 4, 2003
                      '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 10 of 26 , Mar 4, 2003
                        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 11 of 26 , Mar 4, 2003
                          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 12 of 26 , Mar 4, 2003
                            --- 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 13 of 26 , Mar 4, 2003
                              --- 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 14 of 26 , Mar 4, 2003
                                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 15 of 26 , Mar 4, 2003
                                  --- 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 16 of 26 , Mar 4, 2003
                                    --- "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 17 of 26 , Mar 4, 2003
                                      --- 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 18 of 26 , Mar 4, 2003
                                        > 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 19 of 26 , Mar 5, 2003
                                          --- "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 20 of 26 , Mar 5, 2003
                                            '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/
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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 21 of 26 , Mar 5, 2003
                                              --- 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


                                              =====
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                                              would do, sources said: 'The rest is just gesture politics." -- Hoon

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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 22 of 26 , Mar 5, 2003
                                                --- 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

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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 23 of 26 , Mar 5, 2003
                                                  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 24 of 26 , Mar 5, 2003
                                                    --- 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)


                                                    =====
                                                    "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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                                                  • 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 25 of 26 , Mar 5, 2003
                                                      --- 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 26 of 26 , Mar 31, 2003
                                                        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/
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