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Re: [PrimeNumbers] Most prime 2^n+k series

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  • Phil Carmody
    ... Robert and I have put a lot of time and effort into this, so our current records are quite impressive, however, just in the last few days I ve over doubled
    Message 1 of 2 , Mar 3, 2003
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      --- Gary Chaffey <garychaffey@...> wrote:
      > As a parallel project to Robert and Phil's k.2^n+1
      > search I have been looking at PRP series of the form
      > 2^n+k.
      > I have now taken my first promising k value upto
      > n=100000 and have found for k=994218225 there are 118
      > PRPs for n<100000. I think this record is breakable
      > and I am currently testing another value which for
      > n<40600 there are 110 PRPs.

      Robert and I have put a lot of time and effort into this, so our current
      records are quite impressive, however, just in the last few days I've
      over doubled my pre-processing stage's yield, and so we are expecting,
      or at least hoping, to beat our own records in the coming weeks/months.

      118 p from 17136 n (compare Jack's fave, k=577294575, 104/20000)
      129 p from 38278 n
      139 p from 96431 n
      (not all from the same k).

      > If anybody is interested
      > in my results then I will be happy to email them
      > (offlist).

      I assume they'll go on Henri's PRP list.

      > One thing I think is worth noting, is that these
      > series seem to be quite well behaved and back at the
      > beginning of December I predicted in an email to Phil
      > that for k=994218225 I would find 117-119 PRPs.

      Predictions, for these numbers, are funny things...

      I've noticed some very weird-looking behaviour from some of the numbers
      I've tested. From discussions with Robert I think the numbers he's
      picked have had similar behaviour too. Basically we've got spoons!

      http://fatphil.org/maths/prothrace/

      Basically, the numbers start quite densely composite, as k is quite
      large (10^20-10^30 typically). So there's an initial flat zone.

      However, there's a multi-level sieving operation which throws out poor to
      middling potential candidates, and only passes real good'uns. That forces
      the candidates to have a real spurt to meet the very demanding targets, so
      there's an upturn.

      Eventually, when Robert decides that there's one that's worth pushing a long
      way (I pre-filter, and don't really do much testing, I give them to Robert
      to do the middle and upper range tests), its behaviour quite often levels
      off to the slope that its proth weight would suggest, so part 3 of the graph
      is angled at somewhere between the initial flatness and the middle boom.

      However, even this long-term behaviour deviates from the expected density
      quite impressively. For example my favourite candidate has had 2 big boosts
      in the last few weeks, which are clearly visible in the top right of the
      graph above. However, the flat patch between the two bursts does make teh
      whole thing average out in the end.


      Gary, if you want to try the following numbers at 2^n+k and they look
      useful, then you can have my sieving output. They're selected purely on
      their k.2^n+1 behaviour, which is correlated to their 2^n+k behaviour, but
      not necessarily enough for them to be useful to you.

      p/n k/40755
      45/500 14117461 (so k=14117461*40755=575357123055)
      45/500 59179429
      47/500 147584529
      44/500 170838081
      46/500 189810965
      44/500 821346345
      45/500 1082888591
      45/500 1291064811
      44/500 1334615955
      44/500 1348990679
      45/500 1350868261
      44/500 7340655
      44/500 16246853
      45/500 292328043
      44/500 443733409
      45/500 539824005
      44/500 959077665
      44/500 1634788225
      48/500 34705203
      44/500 429687147
      46/500 807332933
      50/500 837188129
      47/500 925976409
      46/500 1249522653
      46/500 1595359025
      44/500 1619433677
      44/500 1644028359
      44/500 1915763193
      44/500 2336275873

      I have tens of thousands where they came from.
      If they're not any good, then a special sieve will need to be written.

      Phil



      =====
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