> As a parallel project to Robert and Phil's k.2^n+1

Robert 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

> 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.

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

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

> in my results then I will be happy to email them

> (offlist).

> One thing I think is worth noting, is that these

Predictions, 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!

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

=====

"Only an admission that he does possess weapons of mass destruction

would do, sources said: 'The rest is just gesture politics." -- Hoon

"Are you still bombing your wife?" -- Winjer

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