Re: [PrimeNumbers] Most prime 2^n+k series
- --- Gary Chaffey <garychaffey@...> wrote:
> As a parallel project to Robert and Phil's k.2^n+1Robert and I have put a lot of time and effort into this, so our current
> search I have been looking at PRP series of the form
> 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.
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 interestedI assume they'll go on Henri's PRP list.
> in my results then I will be happy to email them
> One thing I think is worth noting, is that thesePredictions, for these numbers, are funny things...
> 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.
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!
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.
45/500 14117461 (so k=14117461*40755=575357123055)
I have tens of thousands where they came from.
If they're not any good, then a special sieve will need to be written.
"Only an admission that he does possess weapons of mass destruction
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