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Re: [PrimeNumbers] Eleven successive primes not full period - is this a record?

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  • Jack Brennen
    A few minutes with PARI/GP yields 25 consecutive such primes from 461413 to 461803.
    Message 1 of 12 , Aug 7, 2009
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      A few minutes with PARI/GP yields 25 consecutive such primes from 461413 to 461803.

      julienbenney wrote:
      > One thing I had discovered earlier but just rediscovered today relates to the sequence of primes:
      >
      > 3037, 3041, 3049, 3061, 3067, 3079, 3083, 3089, 3109, 3119, 3121
      >
      > These are eleven consecutive primes not one of which is a full period prime.
      >
      > The question I want to ask is whether there is any other sequence of consecutive non-full-period primes equal to or greater than this sequence. Since it actually runs from the 435th to the 445th prime, it seems all the more suprising and I have never seen it noted.
      >
      >
      >
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    • David Broadhurst
      ... [25, 461413, 461803] [29, 6754837, 6755267] [30, 27214597, 27215039] David
      Message 2 of 12 , Aug 8, 2009
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        --- In primenumbers@yahoogroups.com,
        Jack Brennen <jfb@...> wrote:

        > A few minutes with PARI/GP yields 25 consecutive such primes
        > from 461413 to 461803.

        [25, 461413, 461803]
        [29, 6754837, 6755267]
        [30, 27214597, 27215039]

        David
      • Jens Kruse Andersen
        ... This is trivial as others have shown. The next case of eleven is as small as 4481 to 4561. Consecutive primes which *are* full period primes appear more
        Message 3 of 12 , Aug 8, 2009
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          julienbenney wrote:
          > 3037, 3041, 3049, 3061, 3067, 3079, 3083, 3089, 3109, 3119, 3121
          >
          > These are eleven consecutive primes not one of which is a full period
          > prime.
          >
          > The question I want to ask is whether there is any other sequence of
          > consecutive non-full-period primes equal to or greater than this sequence.

          This is trivial as others have shown.
          The next case of eleven is as small as 4481 to 4561.
          Consecutive primes which *are* full period primes appear more interesting.
          In http://www.primepuzzles.net/puzzles/puzz_095.htm I found 23 from
          239651440411 to 239651440949.

          --
          Jens Kruse Andersen
        • David Broadhurst
          ... Those are much easier to handle, since one may use a kronecker filter, as Chris Nash observed. I prefer Julien s twist: 10 is not a primitive root of unity
          Message 4 of 12 , Aug 8, 2009
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            --- In primenumbers@yahoogroups.com,
            "Jens Kruse Andersen" <jens.k.a@...> wrote:

            > Consecutive primes which *are* full period primes
            > appear more interesting.

            Those are much easier to handle, since one may
            use a kronecker filter, as Chris Nash observed.

            I prefer Julien's twist:

            10 is not a primitive root of unity modulo any of
            the 33 consecutive primes from 923125117 to 923125939

            n=33;p=923125117;v=vector(n);
            for(k=1,33,v[k]=(p-1)/znorder(Mod(10,p));p=nextprime(p+1));
            print(v);

            [6, 2, 2, 4, 2, 30, 6, 4, 7, 3, 6, 2, 7, 2, 8, 3,
            2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 12, 3, 2, 4, 4, 7, 7]

            Any advance on 33 consecutive primes of this kind?

            David
          • Jens Kruse Andersen
            ... [35, 1292613521, 1292614321] [36, 2757553987, 2757554983] -- Jens Kruse Andersen
            Message 5 of 12 , Aug 8, 2009
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              David Broadhurst wrote:
              > Any advance on 33 consecutive primes of this kind?

              [35, 1292613521, 1292614321]
              [36, 2757553987, 2757554983]

              --
              Jens Kruse Andersen
            • David Broadhurst
              ... 10 is not a primitive root of unity modulo any of the 35 consecutive primes beginning with 1292613521 n=35;p=1292613521;v=vector(n);
              Message 6 of 12 , Aug 8, 2009
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                --- In primenumbers@yahoogroups.com,
                "David Broadhurst" <d.broadhurst@...> wrote:

                > Any advance on 33 consecutive primes of this kind?

                10 is not a primitive root of unity modulo any of
                the 35 consecutive primes beginning with 1292613521

                n=35;p=1292613521;v=vector(n);
                for(k=1,n,v[k]=(p-1)/znorder(Mod(10,p));p=nextprime(p+1));
                print(v);

                [10, 2, 2, 2, 2, 3, 2, 7, 2, 2, 4, 3, 4, 2, 3, 8, 14, 4,
                2, 6, 2, 6, 2, 2, 3, 12, 2, 2, 3, 21, 2, 2, 3, 3, 10]

                David
              • David Broadhurst
                ... Ah, Jens beat my case with 35, which was to be expected. David
                Message 7 of 12 , Aug 8, 2009
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                  --- In primenumbers@yahoogroups.com,
                  "Jens Kruse Andersen" <jens.k.a@...> wrote:

                  > > Any advance on 33 consecutive primes of this kind?
                  >
                  > [35, 1292613521, 1292614321]
                  > [36, 2757553987, 2757554983]

                  Ah, Jens beat my case with 35, which was to be expected.

                  David
                • David Broadhurst
                  ... [37, 7088772637, 7088772653] David
                  Message 8 of 12 , Aug 8, 2009
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                    --- In primenumbers@yahoogroups.com,
                    "Jens Kruse Andersen" <jens.k.a@> wrote:

                    > [36, 2757553987, 2757554983]

                    [37, 7088772637, 7088772653]

                    David
                  • David Broadhurst
                    ... Whoops! The last of these 37 primes is 7088773519 n=37;p1=7088772637;v=vector(n);p=p1; for(k=1,n,v[k]=(p-1)/znorder(Mod(10,p));p2=p;p=nextprime(p+1));
                    Message 9 of 12 , Aug 8, 2009
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                      --- In primenumbers@yahoogroups.com,
                      "David Broadhurst" <d.broadhurst@...> wrote:

                      > [37, 7088772637, 7088772653]

                      Whoops! The last of these 37 primes is 7088773519

                      n=37;p1=7088772637;v=vector(n);p=p1;
                      for(k=1,n,v[k]=(p-1)/znorder(Mod(10,p));p2=p;p=nextprime(p+1));
                      print([n,p1,p2,v]);

                      [37, 7088772637, 7088773519,
                      [2, 52, 2, 2, 2, 10, 4, 6, 2, 4, 2, 2, 2, 6, 2, 2, 8, 4,
                      3, 2, 2, 3, 3, 60, 3, 2, 4, 35, 3, 4, 4, 3, 2, 25, 2, 2, 2]]

                      With apologies for the typo,

                      David
                    • David Broadhurst
                      ... 10 is not a primitive root of unity modulo any of the 40 consecutive primes beginning with 22588287443 n=40;p1=22588287443; v=vector(n);p=p1;
                      Message 10 of 12 , Aug 8, 2009
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                        --- In primenumbers@yahoogroups.com,
                        "David Broadhurst" <d.broadhurst@...> wrote:

                        > 37, 7088772637, 7088773519

                        10 is not a primitive root of unity modulo any of
                        the 40 consecutive primes beginning with 22588287443

                        n=40;p1=22588287443;
                        v=vector(n);p=p1;
                        for(k=1,n,v[k]=(p-1)/znorder(Mod(10,p));p2=p;p=nextprime(p+1));
                        print([n,p1,p2,v]);

                        [40, 22588287443, 22588288409,
                        [46, 15, 5, 2, 2, 5, 2, 2, 2, 2, 7, 2, 5, 9, 7, 30, 2, 4, 3, 3,
                        3, 4, 3, 3, 2, 6, 2, 9, 2, 16, 2, 2, 2, 2, 2, 6, 48, 5, 2, 2]]

                        David
                      • David Broadhurst
                        ... 10 is not a primitive root of unity modulo any of the 45 consecutive primes beginning with 24124140487 n=45;p1=24124140487; v=vector(n);p=p1;
                        Message 11 of 12 , Aug 8, 2009
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                          --- In primenumbers@yahoogroups.com,
                          "David Broadhurst" <d.broadhurst@...> wrote:

                          > 10 is not a primitive root of unity modulo any of
                          > the 40 consecutive primes beginning with 22588287443

                          10 is not a primitive root of unity modulo any of
                          the 45 consecutive primes beginning with 24124140487

                          n=45;p1=24124140487;
                          v=vector(n);p=p1;
                          for(k=1,n,v[k]=(p-1)/znorder(Mod(10,p));p2=p;p=nextprime(p+1));
                          print([n,p1,p2,v]);

                          [45, 24124140487, 24124141903,
                          [3, 2, 6, 2, 2, 90, 6, 3, 7, 4, 8, 3, 12, 4, 4,
                          2, 2, 9, 20, 6, 7, 2, 4, 2, 36, 42, 2, 2, 2, 2,
                          2, 2, 117, 10, 3, 15, 3, 4, 2, 2, 10, 2, 2, 2, 7]]

                          David
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