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Known pairs vs OEIS A002072

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  • Jim White
      I wonder if the largest pair list at A002072 should include some statement that certain values are smallest known rather than smallest , as the latter
    Message 1 of 4 , Mar 7, 2012
       
      I wonder if the largest pair list at A002072 should
      include some statement that certain values are
      "smallest known" rather than "smallest", as the
      latter assumes the values can be confirmed in
      some rigorous manner.
       
      I haven't done a rigorous Lehmer-method in yonks,
      I can't remember at what stage I started applying
      period bailout, somewhere around p = 97.
       
      p = 97 is, as far as I know, the greatest prime
      for which the results are rigorous, this result
      being quite recent, ie Luca & Najman (2010).
       
      They used a very non-Lehmer method involving
      computing regulators.
       
      It's a very expensive method, thousands of times
      slower than my modified Lehmer (period bailout),
      but probably hundreds of times faster than a
      "rigorous Lehmer" would be.
       
      The results I computed up to p=173 are
      probably correct, although the monster at p=157
      casts reasonable doubt over all subsequent
      results.  By this stage I was not just applying
      period bailout, but also a bailout on the
      number of prime divisors of D.
       
      This added restriction allowed the
      computations to be completed in days rather
      than months, but the probability that I missed
      bigger monsters is accordingly increased.
       
      Having said that, and assuming both A002072 and
      column k=1 of your chain table are "smallest
      known", I do have a new entry for p = 227, having
      found this pair (S, S+1):
       
      S = 15487655655079919751646464

      This was found via a PTE match at
       Q = 395684061 {0, 3, 3}, {1, 1, 4}
       
      Thus S = Q(Q+3)^2 / 4
       

      Jim White
      Canberra 

      ________________________________
      From: Andrey Kulsha <andrey_601@...>
      To: Jim White <mathimagics@...>
      Sent: Wednesday, 7 March 2012, 4:29
      Subject: Re: [PrimeNumbers] Re: 13-chains of consecutive smooth numbers


      
      Any chance of a text export?
          Here we are: http://www.primefan.ru/stuff/math/maxs.txt

          Chain length from 6 to
      16, N up to 10^13.

          Best regards,

          Andrey

      [Non-text portions of this message have been removed]
    • Jim White
      I m sorry, I said 227, I really meant 229! ________________________________ From: Jim White  Subject: [PrimeNumbers] Known pairs vs
      Message 2 of 4 , Mar 7, 2012
        I'm sorry, I said 227, I really meant 229!



        ________________________________
        From: Jim White <mathimagics@...>
         Subject: [PrimeNumbers] Known pairs vs OEIS A002072


         

        Having said that, and assuming both A002072 and
        column k=1 of your chain table are "smallest
        known", I do have a new entry for p = 227, having
        found this pair (S, S+1):
         
        S = 15487655655079919751646464

        This was found via a PTE match at
         Q = 395684061 {0, 3, 3}, {1, 1, 4}
         
        Thus S = Q(Q+3)^2 / 4
         

        Jim White
        Canberra 

        ________________________________
        From: Andrey Kulsha <andrey_601@...>
        To: Jim White <mathimagics@...>
        Sent: Wednesday, 7 March 2012, 4:29
        Subject: Re: [PrimeNumbers] Re: 13-chains of consecutive smooth numbers


        
        Any chance of a text export?
            Here we are: http://www.primefan.ru/stuff/math/maxs.txt

            Chain length from 6 to
        16, N up to 10^13.

            Best regards,

            Andrey

        [Non-text portions of this message have been removed]




        [Non-text portions of this message have been removed]
      • Andrey Kulsha
        ... These values are indeed the largest known (there may be larger ones). Doubt is not about smallest but about for all i m . ... For p = 229 we have
        Message 3 of 4 , Mar 10, 2012
          > I wonder if the largest pair list at A002072 should
          > include some statement that certain values are
          > "smallest known" rather than "smallest", as the
          > latter assumes the values can be confirmed in
          > some rigorous manner.

          These values are indeed the largest known
          (there may be larger ones). Doubt is not about
          "smallest" but about "for all i>m".

          > I do have a new entry for p = 229, having
          > found this pair (S, S+1):
          >
          > S = 15487655655079919751646464

          For p = 229 we have 5.2*sqrt(p)-7.7 = 71,
          so I expect N about exp(71), which is much
          larger than your N about exp(58).

          Best regards,

          Andrey
        • Jim White
          This 181-smooth pair popped up in a random search:   S = 90672220863645734556839376   Factors   S   = 2^4, 3^3, 11, 23, 29, 31, 37^2, 73, 97,       
          Message 4 of 4 , Mar 12, 2012
            This 181-smooth pair popped up in a random search:
             
            S = 90672220863645734556839376
             
            Factors
              S   = 2^4, 3^3, 11, 23, 29, 31, 37^2, 73, 97,
                    139^2, 163, 167, 181
              S-1 = 5^4, 13^3, 17, 19, 43, 47, 67^2, 101^2,
                    113^2, 173

            [Non-text portions of this message have been removed]
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