## k.2^n+/-1 series

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• Does anyone in the group know if any series of the form k.2^n+/-1 has 100+ known primes? Regards Robert Smith
Message 1 of 12 , Apr 6, 2002
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Does anyone in the group know if any series of the form k.2^n+/-1 has
100+ known primes?

Regards

Robert Smith
• ... See Carlos Rivera s web page at http://www.primepuzzles.net/puzzles/puzz_006.htm The information on that page is seriously out-of-date, but shows that 113
Message 2 of 12 , Apr 6, 2002
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robert44444uk wrote:
>
> Does anyone in the group know if any series of the form k.2^n+/-1 has
> 100+ known primes?
>

See Carlos Rivera's web page at

http://www.primepuzzles.net/puzzles/puzz_006.htm

The information on that page is seriously out-of-date, but shows
that 113 primes of the form 577294575*2^n+1 exist with n <= 33772.

I don't have all of my logs handy, but I would guess that there
are roughly 30 more known primes of that form discovered since
Carlos' page was updated (all of them by me, last I checked :-)

That would give somewhere around 140 known primes of that form.
• ... has ... Robert, The number of primes of this form up to a given n depend on the weight of k (see http://brennen.net/primes/ProthWeight.html and
Message 3 of 12 , Apr 7, 2002
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--- In primenumbers@y..., "robert44444uk" <100620.2351@c...> wrote:
> Does anyone in the group know if any series of the form k.2^n+/-1
has
> 100+ known primes?

Robert,

The number of primes of this form up to a given 'n' depend on
the 'weight' of k (see http://brennen.net/primes/ProthWeight.html and
http://www.glasgowg43.freeserve.co.uk/nashdef.htm). And my guess is
that a PrimoProth form would have a pretty high weight, since they
explicitly exclude small prime factors. Here are some results from
Jack's applet:

[k : 577294575, w : 3.8109277425263]
k : 3710369067405, w : 3.632228059618325
k : 100280245065, w : 3.3005203415048
k : 255255, w : 3.2873890549013387
k : 3234846615, w : 3.1315263921732965
k : 111546435, w : 2.9259932105539006
k : 4849845, w : 2.327663281839659

The first is Jack's 577294575, the others are 17#/2, 19#/2, etc.

By the way, it only took about half an hour to add PrimoProths to
NewPGen, all of the tools required were there (the verification
routine took most of the time; to do the work only required changing

Regards,

Paul.
• By the way, it only took about half an hour to add PrimoProths to NewPGen, all of the tools required were there (the verification routine took most of the
Message 4 of 12 , Apr 7, 2002
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By the way, it only took about half an hour to add PrimoProths to
NewPGen, all of the tools required were there (the verification
routine took most of the time; to do the work only required changing

So when the new version of NewPGen will be available?
Regards,

Marcin.

[Non-text portions of this message have been removed]
• ... It should be ready for release very soon. I have been very busy with it recently - as well as the PrimoProth sieves the release also allows larger bitmaps
Message 5 of 12 , Apr 7, 2002
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--- In primenumbers@y..., "Marcin Lipinski" <apollo@a...> wrote:
> So when the new version of NewPGen will be available?

It should be ready for release very soon. I have been very busy with
it recently - as well as the PrimoProth sieves the release also
allows larger bitmaps (up to 485 Mb, don't ask why 485 rather than
512 or 2Gb); much improved performance for small p; the ability to
automatically test with PFGW once the sieving has finished; a scheme
to improve performance for very sparse, wide ranges; a 'Hide' option;
and the ability to use two save files so that you should never lose
your work if the machine crashes.

If you want to beta test it, I'll be sending the latest release
candidate to yourself and Robert.

Regards,

Paul.
• ... I just thought I d add that the highest Proth weight for k
Message 6 of 12 , Apr 7, 2002
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Paul Jobling wrote:

> Here are some results from Jack's applet:
>
> [k : 577294575, w : 3.8109277425263]
> k : 3710369067405, w : 3.632228059618325
> k : 100280245065, w : 3.3005203415048
> k : 255255, w : 3.2873890549013387
> k : 3234846615, w : 3.1315263921732965
> k : 111546435, w : 2.9259932105539006
> k : 4849845, w : 2.327663281839659
>
> The first is Jack's 577294575, the others are 17#/2, 19#/2, etc.

I just thought I'd add that the highest Proth weight for k < 10^9
is the amazing k = 986963835, with a weight of 4.1295...

The reason that 577294575 has more *known* primes is that it is
much richer in primes for low n. For n<1000, k=577294575 has
56 primes, compared to only 43 primes in that range for k=986963835.

Note that Chad Davis (g22) was actively searching for primes of the
form 986963835*2^n+1 as recently as July 2001, but only up to
n around 180332.

I have completely searched for primes of the form 577294575*2^n+1
up to n = 310000, and substantial partial ranges beyond that.
• ... This reminds me of another similar question. What value of k has the most known twin primes of the form (k*2^n-1, k*2^n+1)? I found an example of k with
Message 7 of 12 , Apr 7, 2002
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robert44444uk wrote:
>
> Does anyone in the group know if any series of the form k.2^n+/-1 has
> 100+ known primes?
>

This reminds me of another similar question. What value of k has
the most known twin primes of the form (k*2^n-1, k*2^n+1)?

I found an example of k with 12 known twin prime pairs some years
ago, but I don't have those logs handy... I do remember that all
12 twin prime pairs had fairly small exponents -- I don't think
that n exceeded 100 for any of the twin prime pairs.
• ... Just out of curiosity, are most folks in the community now using PFGW instead of PRP for Proth and K*2^N-1 candidates? I haven t compared the relative
Message 8 of 12 , Apr 7, 2002
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At 12:00 PM 4/7/2002 +0000, Paul Jobling wrote:
>--- In primenumbers@y..., "Marcin Lipinski" <apollo@a...> wrote:
> > So when the new version of NewPGen will be available?
>
>It should be ready for release very soon. I have been very busy with
>it recently - as well as the PrimoProth sieves the release also
>allows larger bitmaps (up to 485 Mb, don't ask why 485 rather than
>512 or 2Gb); much improved performance for small p; the ability to
>automatically test with PFGW once the sieving has finished;

Just out of curiosity, are most folks in the community now using PFGW
instead of PRP for Proth and K*2^N-1 candidates? I haven't compared the
relative speeds in some time, having been busy with Fermat.exe; I should
expect the speeds to be comparable, since both programs use George's v21
libraries.

Nathan, who would use PFGW constantly if it had the ability to minimize to
the system tray
• ... It does, and has done for some time! Look for the Win_Dev releases, it has this ability, and can be set to restart minimized to the tray, so you can just
Message 9 of 12 , Apr 7, 2002
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>
> Nathan, who would use PFGW constantly if it had the ability to minimize to
> the system tray
>

It does, and has done for some time! Look for the Win_Dev releases, it has
this ability, and can be set to restart minimized to the tray, so you can
just dump it and a range on a computer and put a shortcut in the StartUp

On the k.2^n+/-1 numbers PFGW has a considerable advantage over Proth.exe I
believe; Proth is now mainly for generalised Fermat's only.

Michael.
• ... with ... PFGW ... compared the ... should ... George s v21 ... minimize to ... It has had that ability for about a year now Nathan. It is called
Message 10 of 12 , Apr 7, 2002
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--- In primenumbers@y..., Nathan Russell <nrussell@a...> wrote:
> At 12:00 PM 4/7/2002 +0000, Paul Jobling wrote:
> >--- In primenumbers@y..., "Marcin Lipinski" <apollo@a...> wrote:
> > > So when the new version of NewPGen will be available?
> >
> >It should be ready for release very soon. I have been very busy
with
> >it recently - as well as the PrimoProth sieves the release also
> >allows larger bitmaps (up to 485 Mb, don't ask why 485 rather than
> >512 or 2Gb); much improved performance for small p; the ability to
> >automatically test with PFGW once the sieving has finished;
>
> Just out of curiosity, are most folks in the community now using
PFGW
> instead of PRP for Proth and K*2^N-1 candidates? I haven't
compared the
> relative speeds in some time, having been busy with Fermat.exe; I
should
> expect the speeds to be comparable, since both programs use
George's v21
> libraries.
>
> Nathan, who would use PFGW constantly if it had the ability to
minimize to
> the system tray

It has had that ability for about a year now Nathan. It is called
WinPFGW.exe. BTW, there was a new dev release done just today, which
is the first public release with the v22 Woltman libs. This release
can be found in the Yahoo groups "primeform" files folder.

Jim.
• ... Following up to my previous message: k == 202507305 == 3.5.7.11.13.13487 (k*2^n-1, k*2^n+1) are both prime for these n: 2, 12, 17, 28, 31, 33 42, 55, 62,
Message 11 of 12 , Apr 8, 2002
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--- In primenumbers@y..., Jack Brennen <jack@b...> wrote:
> I found an example of k with 12 known twin prime pairs some years
> ago, but I don't have those logs handy... I do remember that all
> 12 twin prime pairs had fairly small exponents -- I don't think
> that n exceeded 100 for any of the twin prime pairs.

Following up to my previous message:

k == 202507305 == 3.5.7.11.13.13487

(k*2^n-1, k*2^n+1) are both prime for these n:

2, 12, 17, 28, 31, 33
42, 55, 62, 86, 89, 91

[and probably for no other values of n...]
• A belated reply here, but check my webpage www.glasgowg43.freeserve.co.uk/nashprim.htm for straightforwardly-generated numbers with very high Proth weight - in
Message 12 of 12 , May 3, 2002
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A belated reply here, but check my webpage www.glasgowg43.freeserve.co.uk/nashprim.htm
for straightforwardly-generated numbers with very high Proth weight - in the equivalent context of Nash weight.

In terms of searching through those in the webage, some of the k-values are already taken, including:

986963835
302627325
302442855
806586495

(the last by me). There will be plenty of others available which should provide large numbers of small primes of the form k*2^n+1.

If you want to know more about the divisibility properties of these numbers, then go to
www.glasgowg43.freeserve.co.uk/pfaq6.htm

Joe.

-----Original Message-----
From: robert44444uk <100620.2351@...>
Date: 08 April 2002 02:33

On the question of proth weights for k, the highest I found to date
is for 39.37#+2.31#+20.29# (289939302102450) which has a weight of
4.216. I'm sure there are higher values than this.

I am not 100% convinced by the proth weight concept. I ran this
number up to n= circa 3,200 both plus and minus one, and got 43 and
31 primes, which seems pretty average to me. As a comparison,
67#.2^n+1 plus yielded 70 primes up to n=3,300, (and 88 up to
n=18,000) with a proth weight of (only) 4.191

That said, Jack's applet is an interesting tool and I use it!

Carlos Rivera has rejected 23#.2^n-1 as the most populous -1 series,
(95 primes) because it has an index p/ln(n) of 8.00, compared to
Jack's record of 9.083. It seems you must beat not only the absolute
number of primes, but also the index to get a listing for puzzle 6
(see http://www.primepuzzles.net/puzzles/puzz_006.htm)

On processing speed - on my machine, PRP and pfgw are about the same.

Regards

Robert Smith

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