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Most prime 2^n+k series

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  • Gary Chaffey
    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
    Message 1 of 2 , Mar 3, 2003
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      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. If anybody is interested
      in my results then I will be happy to email them
      (offlist).
      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.
      Gary Chaffey

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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 2 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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