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Re: [PrimeNumbers] Nash weights

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  • joe.mclean@it.glasgow.gov.uk
    Phil (et al) I ve done (mostly in the past to be sure) a lot of work on Nash weights, and adjusted Nash weights. Check out :
    Message 1 of 6 , Mar 12, 2001
      Phil (et al)

      I've done (mostly in the past to be sure) a lot of work on Nash
      weights, and adjusted Nash weights. Check out :

      www.glasgowg43.freeserve.co.uk/nashdef.htm

      for definitions

      www.glasgowg43.freeserve.co.uk/nashprim.htm

      for high Nash weights

      www.glasgowg43.freeserve.co.uk/siernash.htm

      for low Nash weights

      I have C programs that calculated adjusted or capped Nash weights for
      k in ranges and with particular modular forms. Capping just involves
      putting a limit on the primes as well as the exponents.

      Joe.


      ______________________________ Reply Separator _________________________________
      Subject: [PrimeNumbers] Nash weights
      Author: Phil Carmody <fatphil@...> at Internet
      Date: 11/03/01 04:21


      Can the resident 'Nash weight calculation' expert please step forward!

      OK, there are 2 factors involved.
      1) absolute exclusion of primes due to explicit factors
      e.g. 27*2^n+/-1 will never be divisible by 3
      2) being not in the right orbit for other primes
      e.g. 27*2^n-1 will only be 4,2,5 mod 7 and 27*2^n+1 will only be 6,4,0 mod 7.
      Therefore 7 helps increase prime density for 27..-, but reduces prime density
      for 27..+ (it removes 1 in 3 rather than 1 in 7).

      Have Nash weights been calculated _in bulk_. I mean real bulk, up to hundreds of
      millions?

      The reason I ask is that I've seen the applet which calculates the weights for a
      particular number, but it struck me that they can be calculated en mass like a
      sieve? Has anyone done this?

      Phil

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    • joe.mclean@it.glasgow.gov.uk
      Hello all. k = 138847 has a standard (negative) Nash weight of 29, which is very respectable for a reasonably small number. The lowest positive Nash weight I
      Message 2 of 6 , Mar 12, 2001
        Hello all.

        k = 138847 has a standard (negative) Nash weight of 29, which is very
        respectable for a reasonably small number. The lowest positive Nash
        weight I found was 9 (yes, nine), so I'm sure that I could find
        something for the negative side in the low teens if I put my mind to
        it, though this would take some time with my current computers.

        However, if someone could write a very fast program for me, that would
        help. How about that Phil. Spookily, I was actually thinking about
        asking for this just last week. Great minds think alike. The logic is
        dead easy, honest.

        Joe.


        ______________________________ Reply Separator _________________________________
        Subject: [PrimeNumbers] Re: Nash weights
        Author: d.broadhurst@... at Internet
        Date: 12/03/01 01:38


        Jack Brennen wrote:

        > I have done some extensive calculations looking for high
        > Proth weights for both k*2^n+1 and k*2^n-1

        I like low weights, too.

        They have been studied for k*2^n+1.
        But in the Riesel case, k*2^n-1,
        I do not know of any lower non-zero
        weight than that for k=138847.

        138847*2^n-1 is prime for n=33 and for no other n<380,000.

        Is there any lower non-zero Nash weight with k<10^6,
        for k*2^n +/- 1 ?

        David



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      • Phil Carmody
        ... I find the measurements to be bizarre, in my mind I want a fractional value, and all these integers are out of range, so to speak. I guess that they re
        Message 3 of 6 , Mar 12, 2001
          On Mon, 12 March 2001, joe.mclean@... wrote:
          > k = 138847 has a standard (negative) Nash weight of 29, which is very
          > respectable for a reasonably small number. The lowest positive Nash
          > weight I found was 9 (yes, nine), so I'm sure that I could find
          > something for the negative side in the low teens if I put my mind to
          > it, though this would take some time with my current computers.

          I find the measurements to be bizarre, in my mind I want a fractional value, and all these integers are out of range, so to speak. I guess that they're scaled, but why should they be?
          What is the "average" Nash weight? (not specifying which average, probably geometric). And what if this is normalised to 1?

          > However, if someone could write a very fast program for me, that would
          > help. How about that Phil. Spookily, I was actually thinking about
          > asking for this just last week. Great minds think alike. The logic is
          > dead easy, honest.

          Jack sent me a mail with a third modifier in it, independent of my first two modifiers. This was to do with (Aurefeulian?) factorisations of prime-power factors of k.

          I believe that if I have the definitive expression for a single number, then I can turn that into a (windowed) sieve technique. Who knows, I may even be able to write some "very fast code" for it. It appears to be a O(sqrt(N)) space, O(N) time algorithm, where N is the number itself, not the number of digits.

          Code like this must already have been written. I'm sure it;s easier for me to tweak than it is to start from scratch?

          Phil

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          Support Eric Weisstein, see http://mathworld.wolfram.com
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