Loading ...
Sorry, an error occurred while loading the content.
 

AP9 record

Expand Messages
  • mikeoakes2@aol.com
    Here is a new AP9 record at 401 digits:- (805227062+54790161*n)*941#+1 is prime for n=0..8 Another slightly smaller AP9 was found a little earlier (see
    Message 1 of 5 , Dec 28, 2006
      Here is a new AP9 record at 401 digits:-
      (805227062+54790161*n)*941#+1 is prime for n=0..8

      Another slightly smaller AP9 was found a little earlier (see below):-
      (434578610+97913159*n)*941#+1 is prime for n=0..8

      All confirmed prime with PFGW -tc

      Input/output statistics:-
      Numbers tested by NewPGen: 1.5*10^9
      NewPGen reduced these (by sieving to 1.0T) to: 3.74*10^8
      Numbers tested by PFGW: 3.0*10^8
      PRP's found by PFGW: 1.6*10^7
      AP8's found: 95
      AP9's found: 2

      Run-time statistics:-
      NewPGen: 7 GHz-days
      PFGW: 87 GHz-days
      Pascal program to find AP's: 11 GHz-days

      Remarkably, both AP9's were actually found by manually extending the 14
      AP8's whose 9th term was outside the range examined by the AP-detector.
      The slightly smaller one was found a few msecs earlier, so was for that
      length of time the record holder.
      (Does that qualify it for a listing on your nice page, Jens?:-)

      The number of AP8's is about as expected to find one AP9 - the second
      one must have been popped in by Santa Klaus.

      -Mike Oakes
    • David Broadhurst
      ... Congrats. Also to Ken for an AP7: http://primes.utm.edu/primes/page.php?id=79092 David Note to Chris: Ken s will need a space adding to the comment if it
      Message 2 of 5 , Dec 28, 2006
        --- In primeform@yahoogroups.com, mikeoakes2@... wrote:
        >
        > Here is a new AP9 record at 401 digits:-
        > (805227062+54790161*n)*941#+1 is prime for n=0..8

        Congrats. Also to Ken for an AP7:

        http://primes.utm.edu/primes/page.php?id=79092

        David

        Note to Chris: Ken's will need a space adding to
        the comment if it is to get archived as an AP.
      • Chris Caldwell
        ... Let me echo David, and others, congratulations! Nice job. (I fixed the comments) CC
        Message 3 of 5 , Dec 28, 2006
          > Congrats. Also to Ken for an AP7:
          >
          > http://primes.utm.edu/primes/page.php?id=79092

          Let me echo David, and others, congratulations! Nice job. (I fixed the comments)

          CC
        • Jens Kruse Andersen
          ... Congratulations on another good AP record! That put you ahead: Beating my AP9 changed our AP score from 5-5 to 6-4 in your favour. Markus Frind also has 4.
          Message 4 of 5 , Dec 28, 2006
            Mike Oakes wrote:
            > Here is a new AP9 record at 401 digits:-
            > (805227062+54790161*n)*941#+1 is prime for n=0..8
            >
            > Another slightly smaller AP9 was found a little earlier
            > (see below):-
            > (434578610+97913159*n)*941#+1 is prime for n=0..8
            >
            > The slightly smaller one was found a few msecs earlier, so was
            > for that length of time the record holder.
            > (Does that qualify it for a listing on your nice page, Jens?:-)

            Congratulations on another good AP record!
            That put you ahead: Beating my AP9 changed our AP score from 5-5
            to 6-4 in your favour. Markus Frind also has 4. And an hour ago
            Jaroslaw Wroblewski beat my AP20, so now I have 3 (same as Ken
            Davis). I started 2006 with 11 (and was back at 11 six months ago),
            but it's nice to see my page is creating attention for subtitanic
            AP's.

            http://hjem.get2net.dk/jka/math/aprecords.htm is updated (twice).

            The two AP9 discoveries were so close that I only list the largest.
            My page only allows proven primes, so if I did consider both AP9 then
            the primality proofs would determine it (assuming the PRP AP's were
            found before the proofs). But should computer testing order, or human
            reading order of computer output, be decisive if they are different?
            Chris chose reading order at http://primes.utm.edu/notes/by_year.html
            It's also discussed at
            http://www.mersenneforum.org/showthread.php?p=68387

            Ken Davis has now found four AP7 for k*3011#+1. The latest doesn't
            have the largest prime, so my page doesn't list it.

            The recent AP records by Jaroslaw Wroblewski have an interesting form:
            AP20: 9372688136871853 + 43#*n, n = 0..19
            AP20: 11735227242889999 + 43#*n, n = 0..19
            AP18: 398182070391807316627 + 53#*n, n = 0..17
            He searched long AP's with exactly those progressions, so the
            algorithm was "search simultaneous primes with fixed pattern" and not
            the usual "compute a prime pool and test it for AP's".

            The used primorial 43# is too large to qualify his two AP20 as the
            first allowed cases of more than 18 simultaneous primes at
            http://hjem.get2net.dk/jka/math/simultprime.htm
            (His AP submissions didn't mention this page or simultaneous primes)
            A reason to disallow AP's with small primes and so large primorial
            progression: The smallest "simultaneous" cases become far smaller
            than tuplets, Cunningham chains and most other forms (smallest 18-
            tuplet above 100 has 25 digits). Any number p without a prime factor
            <= 43 is an admissible start for an AP20 (or an AP43) with
            progression 43#, since p + 43#*n hever has a factor <= 43 if p
            doesn't.

            "20 consecutive numbers without a prime factor <= 43 are all prime"
            should clearly not be allowed as 20 "simultaneous" primes (why stop
            at 43 if this was allowed?). But it's a judgement call when to allow
            an AP which avoids lots of small factors in a more "natural" way. The
            rules have not changed since the page was created, but they are not
            set in stone.

            --
            Jens Kruse Andersen
          • mikeoakes2@aol.com
            Jens Kruse Andersen wrote:- ... But you have a terrifically good one: that AP15: nearly 2 years it has stood, and I m not about to take it on. Quality is an
            Message 5 of 5 , Dec 28, 2006
              Jens Kruse Andersen wrote:-

              >And an hour ago Jaroslaw Wroblewski beat my AP20,
              >so now I have 3 (same as Ken Davis).
              >

              But you have a terrifically good one: that AP15: nearly 2 years it has
              stood, and I'm not about to take it on.
              Quality is an important as quantity!

              And it was your milestone post
              http://tech.groups.yahoo.com/group/primenumbers/message/15965
              that initiated my own (and I guess several others') interest in the
              subject.

              Best regards

              Mike
            Your message has been successfully submitted and would be delivered to recipients shortly.