## (n-1)*2^n+1 not often prime?

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• While doing some aimless playing around with Proth-like sequences, I noticed that: (n+1)*2^n+1 seems to generate many more primes than: (n-1)*2^n+1 For n in
Message 1 of 6 , Jan 6, 2003
While doing some aimless playing around with Proth-like sequences,
I noticed that:

(n+1)*2^n+1

seems to generate many more primes than:

(n-1)*2^n+1

For n in [0..999], the first sequence yields 16 primes, the
second only yields 7 primes.

Hmmm... Why is that? Can the Proth weight concept be extended to
these sequences? Which values of C give many primes of the form
(n+C)*2^n+1? Are there any values of C for which there are no
primes at all of the form (n+C)*2^n+1?

Note that C=0 yields the sequence n*2^n+1, the Cullen numbers,
and that primes in this sequence seem to be quite sparse.
• ... All three forms I think have exactly the same sieving properties. If they were to deviate from each other s densities wildly, that would turn everything I
Message 2 of 6 , Jan 6, 2003
--- Jack Brennen <jack@...> wrote:
> While doing some aimless playing around with Proth-like sequences,
> I noticed that:
>
> (n+1)*2^n+1
>
> seems to generate many more primes than:
>
> (n-1)*2^n+1
>
> For n in [0..999], the first sequence yields 16 primes, the
> second only yields 7 primes.
>
>
> Hmmm... Why is that? Can the Proth weight concept be extended to
> these sequences? Which values of C give many primes of the form
> (n+C)*2^n+1? Are there any values of C for which there are no
> primes at all of the form (n+C)*2^n+1?
>
> Note that C=0 yields the sequence n*2^n+1, the Cullen numbers,
> and that primes in this sequence seem to be quite sparse.

All three forms I think have exactly the same sieving properties.
If they were to deviate from each other's densities wildly, that would turn

e.g. look at p=7
let the pair <a,b,c> be { (n+C)*2^n+1 s.t. n==a(mod 7), n==b(mod 3), C=c }

Clearly <a,b,x> = <a+x,b,0>.

I'll start by just plucking a few arbitrary looking example values.
C=-1 has <6,2,-1>%7==0, <0,0,-1>%7==0
C=0 has <5,2,0> %7==0, <6,0,0> %7==0
C=+1 has <4,2,1> %7==0, <5,3,1> %7==0

Lemma:
<a,b,x> == <a+6y,b+6y,x+y> (mod 7)

Proof of lemma:
<a+6y,b+6y,x+y> = { (n+y+x)*2^n+1 s.t. n==a+6y (mod 7), n==b+6y (mod 3) }
= { (n+y+x)*2^n+1 s.t. n==a-y (mod 7), n==b (mod 3) }
= <a-y,b,y+x>
= <a,b,c>
[]

Corollary (who needs a theorem?)
All changing C does is rotate the residues between different n values.

Generalise for all primes, and you've proved that they all have the same
limiting density.

Phil

=====
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Sex and death. -- Ian 'Lemmy' Kilminster

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• ... that is. ===== The answer to life s mystery is simple and direct: Sex and death. -- Ian Lemmy Kilminster
Message 3 of 6 , Jan 6, 2003
--- Phil Carmody <thefatphil@...> wrote:
> = <a,b,c>

<a,b,x> that is.

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Sex and death. -- Ian 'Lemmy' Kilminster

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• In a message dated 06/01/03 08:32:49 GMT Standard Time, jack@brennen.net ... Jack: your second form is treated in my posting of 30 Dec 2000 Gaussian analogues
Message 4 of 6 , Jan 6, 2003
In a message dated 06/01/03 08:32:49 GMT Standard Time, jack@...
writes:

> While doing some aimless playing around with Proth-like sequences,
> I noticed that:
>
> (n+1)*2^n+1
>
> seems to generate many more primes than:
>
> (n-1)*2^n+1
>
> For n in [0..999], the first sequence yields 16 primes, the
> second only yields 7 primes.
>
>

Jack: your second form is treated in my posting of 30 Dec 2000 "Gaussian
analogues of the Cullen and Woodall primes":-

Mike

[Non-text portions of this message have been removed]
• I did some tests with (n+C)*2^n+1 and (n+C)^n-1 in the interval [10000,11000] : == C=1 1 prp 0 prp C=-1
Message 5 of 6 , Jan 6, 2003
I did some tests with (n+C)*2^n+1 and (n+C)^n-1 in the interval
[10000,11000] :

==> C=1 1 prp 0 prp
C=-1 1 prp 1 prp
C=3 0 prp 1 prp
C=-3 1 prp 0 prp

distributed as expected , I think

could you call them for C=1 near Cullen resp. near Woodall ?

rob

----- Original Message -----
From: Jack Brennen <jack@...>
Sent: Monday, January 06, 2003 9:32 AM
Subject: [PrimeNumbers] (n-1)*2^n+1 not often prime?

> While doing some aimless playing around with Proth-like sequences,
> I noticed that:
>
> (n+1)*2^n+1
>
> seems to generate many more primes than:
>
> (n-1)*2^n+1
>
> For n in [0..999], the first sequence yields 16 primes, the
> second only yields 7 primes.
>
>
> Hmmm... Why is that? Can the Proth weight concept be extended to
> these sequences? Which values of C give many primes of the form
> (n+C)*2^n+1? Are there any values of C for which there are no
> primes at all of the form (n+C)*2^n+1?
>
> Note that C=0 yields the sequence n*2^n+1, the Cullen numbers,
> and that primes in this sequence seem to be quite sparse.
>
> Unsubscribe by an email to: primenumbers-unsubscribe@yahoogroups.com
> The Prime Pages : http://www.primepages.org/
>
>
>
> Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/
>
>
• Rob Binnecamp wrote: I did some tests with (n+C)*2^n+1 and (n+C)^n-1 in the interval [10000,11000] : == C=1 1 prp
Message 6 of 6 , Jan 6, 2003
Rob Binnecamp wrote:

"I did some tests with (n+C)*2^n+1 and (n+C)^n-1 in the interval
[10000,11000] :

==> C=1 1 prp 0 prp
C=-1 1 prp 1 prp
C=3 0 prp 1 prp
C=-3 1 prp 0 prp

distributed as expected , I think

could you call them for C=1 near Cullen resp. near Woodall ?"

That's what I call them & here's my results:

Near-Cullen & Near-Woodall primes

form (k+/-1)*2^k+/-1

(1+1)*2^1-1
(2+1)*2^2-1
(3+1)*2^3-1
(4+1)*2^4-1
(5+1)*2^5-1
(9+1)*2^9-1
(14+1)*2^14-1
(15+1)*2^15-1
(16+1)*2^16-1
(27+1)*2^27-1
(45+1)*2^45-1
(122+1)*2^122-1
(125+1)*2^125-1
(213+1)*2^213-1
(242+1)*2^242-1
(256+1)*2^256-1
(263+1)*2^263-1
(290+1)*2^290-1
(855+1)*2^855-1
(1059+1)*2^1059-1
(2273+1)*2^2273-1
(3945+1)*2^3945-1
(3999+1)*2^3999-1
(9512+1)*2^9512-1
(14127+1)*2^14127-1
(16486+1)*2^16486-1
(20056+1)*2^20056-1
(28834+1)*2^28834-1
(41493+1)*2^41493-1(Lifchitz)
(159147+1)*2^159147-1(Ballard) as 79574*4^79574-1
(227139+1)*2^227139-1(Ballard) as 113570*4^113570-1
//[227139]

(2-1)*2^2-1
(4-1)*2^4-1
(5-1)*2^5-1
(11-1)*2^11-1
(28-1)*2^28-1
(35-1)*2^35-1
(235-1)*2^235-1
(325-1)*2^325-1
(551-1)*2^551-1
(688-1)*2^688-1
(7037-1)*2^7037-1
(8896-1)*2^8896-1
(10020-1)*2^10020-1(Mike Oakes)
(13597-1)*2^13597-1(Mike Oakes)
(21347-1)*2^21347-1(Lifchitz)
(118020-1)*2^118020-1(Harvey)
//[145287]

(2+1)*2^2+1
(5+1)*2^5+1
(6+1)*2^6+1
(13+1)*2^13+1
(26+1)*2^26+1
(65+1)*2^65+1 divides GF(63,8)
(66+1)*2^66+1
(86+1)*2^86+1
(114+1)*2^114+1
(133+1)*2^133+1
(186+1)*2^186+1
(294+1)*2^294+1
(445+1)*2^445+1
(866+1)*2^866+1
(1325+1)*2^1325+1
(1478+1)*2^1478+1
(1823+1)*2^1823+1
(2765+1)*2^2765+1
(7553+1)*2^7553+1
(7943+1)*2^7943+1
(10178+1)*2^10178+1
(20960+1)*2^20960+1
(20964+1)*2^20964+1
(21337+1)*2^21337+1
(26562+1)*2^26562+1
(85374+1)*2^85374+1(Lifchitz)
(96749+1)*2^96749+1(Kazuyoshi)
//[134384(odds to 217537)]

(2-1)*2^2+1
(3-1)*2^3+1
(7-1)*2^7+1
(27-1)*2^27+1
(51-1)*2^51+1 divides GF(48,10)
(55-1)*2^55+1
(81-1)*2^81+1
(1471-1)*2^1471+1
(1483-1)*2^1483+1
(8668-1)*2^8668+1(MikeOakes)
(10885-1)*2^10885+1(Lifchitz)
(20803-1)*2^20803+1(MikeOakes)
(32605-1)*2^32605+1(MikeOakes)
(36391-1)*2^36391+1(MikeOakes)
(57004-1)*2^57004+1(Harvey)
(61627-1)*2^61627+1 ""
(88651-1)*2^88651+1(Jo Yeong Uk)
(89731-1)*2^89731+1 ""
(133928-1)*2^133928+1(Harvey)
(153428-1)*2^153428+1(Harvey)
//[200000]

[Non-text portions of this message have been removed]
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