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Triplet record with -x5231887

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  • David Broadhurst
    For this 4135-digit triplet record, Pfgw prompted for the extra bits -x5231887, which was quicker than bridging such a small BLS deficit by ECM effort, or by
    Message 1 of 5 , Sep 26, 2002
      For this 4135-digit triplet record, Pfgw prompted
      for the extra bits -x5231887, which was quicker
      than bridging such a small BLS deficit by ECM effort,
      or by invoking Konyagin-Pomerance.

      Here is the log:

      <<

      Primality testing
      (39553075974*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 5
      [N-1/N+1, Brillhart-Lehmer-Selfridge]
      Calling N+1 BLS with factored part 33.50%
      and helper 0.01% (100.51% proof)
      (39553075974*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 5
      is prime! (82.639000 seconds)

      Primality testing
      (39553075974*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 7
      [N-1/N+1, Brillhart-Lehmer-Selfridge]
      Calling N-1 BLS with factored part 33.50%
      and helper 0.01% (100.51% proof)
      (39553075974*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 7
      is prime! (35.982000 seconds)

      Primality testing
      (39553075974*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 11
      [N-1/N+1, Brillhart-Lehmer-Selfridge]
      Calling N+1 BLS with factored part 33.28%
      and helper 0.01% (99.85% proof)
      Proof incomplete rerun with -x5231887
      (39553075974*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 11
      is Fermat and Lucas PRP! (18.246000 seconds)

      [doing as suggested:]

      Primality testing
      (39553075974*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 11
      [N-1/N+1, Brillhart-Lehmer-Selfridge]
      Calling N+1 BLS with factored part 33.28%
      and helper 0.01% (99.85% proof)

      1/5231887

      8193/5231887

      [snip 600 similar progress reports]

      5210113/5231887

      5218305/5231887

      5226497/5231887
      (39553075974*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 11
      is prime! (579.723000 seconds)

      [and finally here are the proofs of the small
      twins that seed this record triplet:]

      Primality testing 4436*3251#+1
      [N-1/N+1, Brillhart-Lehmer-Selfridge]
      Calling N-1 BLS with factored part 100.00%
      and helper 0.57% (300.59% proof)
      4436*3251#+1 is prime! (6.700000 seconds)

      Primality testing 4436*3251#-1
      [N-1/N+1, Brillhart-Lehmer-Selfridge]
      Calling N+1 BLS with factored part 100.00%
      and helper 0.37% (300.42% proof)
      4436*3251#-1 is prime! (18.717000 seconds)

      >>

      David Broadhurst
    • David Broadhurst
      Hmm. That record lasted only for hours; this 5-7-11 triplet is a mite bigger: Primality testing (90159302514*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 +
      Message 2 of 5 , Sep 27, 2002
        Hmm. That record lasted only for hours;
        this 5-7-11 triplet is a mite bigger:

        Primality testing
        (90159302514*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 5
        [N-1/N+1, Brillhart-Lehmer-Selfridge]
        Calling N+1 BLS with factored part 33.33%
        and helper 0.02% (100.02% proof)
        (90159302514*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 5
        is prime! (71.673000 seconds)

        Primality testing
        (90159302514*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 7
        [N-1/N+1, Brillhart-Lehmer-Selfridge]
        Calling N-1 BLS with factored part 33.33%
        and helper 0.39% (100.39% proof)
        (90159302514*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 7
        is prime! (37.424000 seconds)

        Primality testing
        (90159302514*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 11
        [N-1/N+1, Brillhart-Lehmer-Selfridge]
        Calling N+1 BLS with factored part 33.49%
        and helper 0.03% (100.50% proof)
        (90159302514*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 11
        is prime! (18.306000 seconds)

        with no extra bits needed.

        I still wait for a 7-11-13 triplet, at 4000+ digits,
        to exemplify the cubic Ansatz more fully.

        David Broadhurst
      • David Broadhurst
        I had lively hopes when noting that ... Not long to wait, in fact:
        Message 3 of 5 , Sep 28, 2002
          I had lively hopes when noting that

          > I still wait for a 7-11-13 triplet, at 4000+ digits,
          > to exemplify the cubic Ansatz more fully.

          Not long to wait, in fact:

          <<

          Primality testing
          (18599651274*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 7
          [N-1/N+1, Brillhart-Lehmer-Selfridge]
          Calling N-1 BLS with factored part 33.34%
          and helper 0.25% (100.29% proof)
          (18599651274*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 7
          is prime! (33.098000 seconds)

          Primality testing
          (18599651274*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 11
          [N-1/N+1, Brillhart-Lehmer-Selfridge]
          Calling N+1 BLS with factored part 33.28%
          and helper 0.15% (100.00% proof)
          1/4
          (18599651274*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 11
          is prime! (18.286000 seconds)

          Primality testing
          (18599651274*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 13
          [N-1/N+1, Brillhart-Lehmer-Selfridge]
          Calling N-1 BLS with factored part 33.28%
          and helper 0.04% (99.89% proof)
          Proof incomplete rerun with -x123013
          (18599651274*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 13
          is Fermat and Lucas PRP! (22.422000 seconds)

          [rerun as instructed:]

          Primality testing
          (18599651274*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 13
          [N-1/N+1, Brillhart-Lehmer-Selfridge]
          Calling N-1 BLS with factored part 33.28%
          and helper 0.04% (99.89% proof)
          1/2460259
          8193/2460259
          16385/2460259
          24577/2460259
          32769/2460259
          40961/2460259
          49153/2460259
          57345/2460259
          65537/2460259
          73729/2460259
          81921/2460259
          90113/2460259
          98305/2460259
          106497/2460259
          114689/2460259
          122881/2460259
          (18599651274*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 13
          is prime! (36.102000 seconds)

          >>

          But it's smaller than yesterday's 5-7-11 records...

          David Broadhurst
        • David Broadhurst
          And, for good measure, another 7-11-13 BLS Triplet which is now the current Triplet record holder:
          Message 4 of 5 , Sep 28, 2002
            And, for good measure, another 7-11-13 BLS Triplet
            which is now the current Triplet record holder:

            <<

            Primality testing
            (108748629354*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 7
            [N-1/N+1, Brillhart-Lehmer-Selfridge]
            Calling N-1 BLS with factored part 33.36%
            and helper 0.15% (100.26% proof)
            (108748629354*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 7
            is prime! (32.857000 seconds)

            Primality testing
            (108748629354*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 11
            [N-1/N+1, Brillhart-Lehmer-Selfridge]
            Calling N+1 BLS with factored part 33.28%
            and helper 0.14% (99.96% proof)
            1/94
            (108748629354*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 11
            is prime! (21.782000 seconds)

            Primality testing
            (108748629354*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 13
            [N-1/N+1, Brillhart-Lehmer-Selfridge]
            Calling N-1 BLS with factored part 33.28%
            and helper 0.28% (100.11% proof)
            (108748629354*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 13
            is prime! (21.060000 seconds)

            >>

            Now I should trawl my 1-5-7-11-13 quintuply-sieved
            database, already PFGWed at 7, in case there is a
            record-breaking 1-7-13 CPAP3 lurking there too.
            [Current CPAP3 record is merely 3161 digits.]

            Note that a CPAP3 has a probability only 1/3 of
            of that for finding a triplet. And I was lucky
            to get 4 triplets. No harm in hoping...

            David Broadhurst
          • paulunderwooduk
            ... David , daily -- even hourly -- you are re-writing the top triplet records. Will there be a fifth by sunset? ... Nice work! Add diligence to your luck!
            Message 5 of 5 , Sep 28, 2002
              --- In primeform@y..., "David Broadhurst" <d.broadhurst@o...> wrote:
              > And, for good measure, another 7-11-13 BLS Triplet
              > which is now the current Triplet record holder:
              >
              David , daily -- even hourly -- you are re-writing the top triplet
              records. Will there be a fifth by sunset?
              ...

              > And I was lucky
              > to get 4 triplets. No harm in hoping...

              Nice work! Add diligence to your luck!

              Paul
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