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Re: Some drag-racing progress...

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  • robert44444uk
    ... IMHO, the sprint start will provide the most efficient route to providing the top candidates. I am not certain what the maximum # of primes by n=10,000
    Message 1 of 10 , Oct 29, 2011
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      --- In primenumbers@yahoogroups.com, "djbroadhurst" <d.broadhurst@...> wrote:
      >
      > --- In primenumbers@yahoogroups.com,
      > "robert44444uk" <robert_smith44@> wrote:
      >
      > > The last column refers to the Payam muliplier
      >
      > For Payamic progress on k*10^n-1, see
      > http://homepage2.nifty.com/m_kamada/math/primecount.txt
      > with the record-holder at
      > http://primes.utm.edu/primes/page.php?id=102111#comments
      >
      > Note that this is not drag racing, in its strict sense;
      > Makoto Kamada erects no artificial finishing tape.
      > It makes sense to have a sprint start, as in a drag race.
      > But it's also a good idea to keep k < 2^63, so that you
      > can use NewPgen and PFGW efficiently when the ultra-marathon
      > stage takes over, with n > 200000, fit for Prime Page entry.
      >
      > Best wishes
      >
      > David
      >

      IMHO, the "sprint start" will provide the most efficient route to providing the top candidates.

      I am not certain what the maximum # of primes by n=10,000 is for any k such that k<2^63, but it is possibly about 95 for base 2. This is already 26 primes behind the current leader, and 15 behind the sorts of k that I can find weekly running my "sprint start" software, and 10 behind those candidates which I find every day.

      Regards

      Robert
    • mikeoakes2
      ... Two years ago I did Sierpinski and Riesel drag racing , for both normal and dual forms. I investigated ALL k
      Message 2 of 10 , Oct 30, 2011
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        --- In primenumbers@yahoogroups.com, "robert44444uk" <robert_smith44@...> wrote:
        >
        > IMHO, the "sprint start" will provide the most efficient route to providing the top candidates.
        >
        > I am not certain what the maximum # of primes by n=10,000 is for any k such that k<2^63, but it is possibly about 95 for base 2. This is already 26 primes behind the current leader, and 15 behind the sorts of k that I can find weekly running my "sprint start" software, and 10 behind those candidates which I find every day.
        >
        > Regards
        >
        > Robert
        >

        Two years ago I did Sierpinski and Riesel "drag racing", for both normal and dual forms.
        I investigated ALL k < 10^9, and used the sprint-start performance (up to n=100) to select the best candidates, then to 1000 for the next cut, and so on.
        Here are the winning counts of primes, for three ranges of n.

        Sierpinski k*2^n+1:-
        Best to n=100
        k=232241655=3*5*19*814883 31
        Best to n=1000
        k=101822175=3^2*5^2*7*13*4973 54
        k=902210505=3*5*7*8592481 54
        Best to n=10000
        k=78697515=3*5*13*403577 79
        k=902210505=3*5*7*8592481 79

        Riesel k*2^n-1:-
        Best to n=100
        k=201434145=3*5*11*29*43*89 30
        Best to n=1000
        k=144061665=3*5*11*137*6373 54
        Best to n=10000
        k=80932995=3^2*5*11*13*12577 83

        Dual Sierpinski 2^n+k:-
        Best to n=100
        k=454245=3*5*11*2753 37
        k=130988565=3^2*5*19*23*6661 37
        Best to n=1000
        k=321078615=3*5*11*13*181*827 67
        Best to n=10000
        k=321078615=3*5*11*13*181*827 107
        and continuing this "champion of champions" up to n=100000 gives the exceptionally high count of 134.

        Dual Riesel 2^n-k:-
        Best to n=100
        k=9548565=3*5*13*23*2129 38
        k=688366965=3*5*11*13*269*1193 38
        Best to n=1000
        k=485186295=3*5*11*2940523 69
        Best to n=10000
        k=527175=3^3*5^2*11*71 96

        It is interesting that the ONLY decently performing k without 3 as a factor was the following, with Sierpinski form k*2^n+1:-
        k=922035=5*11*17*1061
        To n=100 19
        To n=1000 50
        To n=10000 69

        It seems that the dual forms tend to have rather higher prime densities.
        I assume this can be accounted for by them being about a factor k smaller.
        Anyone like to quantify this, using PNT?

        Mike
      • robert44444uk
        ... Hi Mike, I would concur duals are slightly more dense. I did not do much on this back in the day, but the following were the best I found on the Sierpinski
        Message 3 of 10 , Oct 30, 2011
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          --- In primenumbers@yahoogroups.com, "mikeoakes2" <mikeoakes2@...> wrote:

          >
          > Dual Sierpinski 2^n+k:-
          > Best to n=100
          > k=454245=3*5*11*2753 37
          > k=130988565=3^2*5*19*23*6661 37
          > Best to n=1000
          > k=321078615=3*5*11*13*181*827 67
          > Best to n=10000
          > k=321078615=3*5*11*13*181*827 107
          > and continuing this "champion of champions" up to n=100000 gives the exceptionally high count of 134.
          >
          > Dual Riesel 2^n-k:-
          > Best to n=100
          > k=9548565=3*5*13*23*2129 38
          > k=688366965=3*5*11*13*269*1193 38
          > Best to n=1000
          > k=485186295=3*5*11*2940523 69
          > Best to n=10000
          > k=527175=3^3*5^2*11*71 96

          >
          > It seems that the dual forms tend to have rather higher prime densities.
          > I assume this can be accounted for by them being about a factor k smaller.
          > Anyone like to quantify this, using PNT?
          >
          > Mike
          >
          Hi Mike, I would concur duals are slightly more dense. I did not do much on this back in the day, but the following were the best I found on the Sierpinski side

          Using M terminology where M(x-1) = multiple of primes up to x from the list 3,5,11,13,19,29,37,53,59,61,67,83,101,107

          k=69338767*M(28) 99 primes by n=5650 or 99/5650
          k=2852139845*M(36) 112/6000
          k=132233299377*M(52)147/40917
          k=19379084737*M(58) 127/20000
          k=2231109458141*M(67) 158/59542
          k=835387103*M(82)56/500
          k=157873677392081*M(106)135/32778

          Best to 100 primes: possibly a shade over n=3000, pretty similar to the Proths: k=21385711867*M(37) had 99/3000, don't have details after that.

          The best performer above, the M(67) candidate compares very favourably to the best at 158 primes for proths of n=164463 for Riesel side and n=95487 for Sierpinski side.

          Best to 1000: 2852139845*M(53) 82/1000, far above Proth best of 73

          Best to 100: 32 by several candidates, behind the Proth best of 34

          All th dual records can be improved significantly. I have checked about 1 trillion (!) k values _ much more work needed for the duals

          regards

          robert
        • robert44444uk
          ... Ugh, keyboard issues. I checked about 1 trillion (!) k values of payam proths, mainly on the Riesel side, but only a few million for the duals.
          Message 4 of 10 , Oct 30, 2011
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            > All th dual records can be improved significantly. I have checked about 1 trillion (!) k values _ much more work needed for the duals
            >
            > regards
            >
            > robert
            >

            Ugh, keyboard issues. I checked about 1 trillion (!) k values of payam proths, mainly on the Riesel side, but only a few million for the duals.
          • robert44444uk
            ... Phil A further advance, on the Riesel side, just failing at 75 primes: 74 972 147707435198851 R 52
            Message 5 of 10 , Dec 14, 2011
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              --- In primenumbers@yahoogroups.com, Phil Carmody <thefatphil@...> wrote:
              >
              > --- On Thu, 10/27/11, robert44444uk <robert_smith44@...> wrote:
              > > Phil Carmody <thefatphil@> wrote:
              > > > Firstly, Jack, can you update Carlos Rivera with your target values,
              > > > otherwise we don't know exactly what we're trying to beat!
              > > >
              > > > In response to Jack's "Any of these milestones are exceptional", with
              > > > reference to 56 primes below 1000, I now have a number with 64 primes
              > > > before n=1000. Is that some kind of record? (Weight=4.786)
              > > > (no 63s, 2 62s, 4 61s, 7 60s, 14 59s, 22 58s, 29 57s, not bad for 24 hours
              > > > work on a 5 year old machine.)
              > > >
              > > > Sure, it's  a 25-digit k, which detracts from the achievement (or
              > > > does it?), but even with numbers of that size I believe that >60
              > > > primes, in particular 64, before n=1000 must still be somewhat of
              > > > a rarity.
              > > >
              > > > I just don't have the 'feel' of how to measure these things yet.
              > >
              > > For the record, the best performing Riesel I have found at n=1000 is
              > > k=19122572047641*3*5*11*13*19*29*37*53 with 73 primes, the 73rd is at n=963
              >
              > Stunning!
              >
              > I don't know how up-to-date my database is, as I didn't fully get the website up and running whilst I was still priming, but the best I have to hand are the following:
              >
              > mysql> select * from records left join candidate on records.cand=candidate.id where n<=1000 && p>=70;
              > +------+----+-----+------+--------+--------------+---------------+------+------+
              > | cand | p | n | id | finder | k | m | plus | dual |
              > +------+----+-----+------+--------+--------------+---------------+------+------+
              > | 16 | 70 | 847 | 16 | g106 | 792030929331 | 2317696095 | 1 | 0 |
              > | 205 | 70 | 956 | 205 | pcrc | 55120464273 | 8341388245905 | -1 | 0 |
              > +------+----+-----+------+--------+--------------+---------------+------+------+
              >
              > Which is odd, as the m for the +1 case is the same small prime multiplier as yours, so my database has its signs all wrong (fortunately an easy fix).
              >
              > I did think that I broke clear of 70, but at the moment I have no proof of that. (And if I did, it was probably only 1 or 2 more.) I don't even know where all my files for that search even went. They were large enough, I may have just binned them :-(
              >
              > Good work, Robert!
              >
              > Phil
              >

              Phil

              A further advance, on the Riesel side, just failing at 75 primes:

              74 972 147707435198851 R 52
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