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

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  • Jack Brennen
    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
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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 [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.
    • Phil Carmody
      ... 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
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        --- 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
        everything I know about sieve theory on its head.

        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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      • Phil Carmody
        ... 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
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          --- Phil Carmody <thefatphil@...> wrote:
          > = <a,b,c>

          <a,b,x> that is.



          =====
          The answer to life's mystery is simple and direct:
          Sex and death. -- Ian 'Lemmy' Kilminster

          __________________________________________________
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        • mikeoakes2@aol.com
          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
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            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":-
            http://groups.yahoo.com/group/primenumbers/message/5

            Mike


            [Non-text portions of this message have been removed]
          • Rob Binnekamp
            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
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              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@...>
              To: <primenumbers@yahoogroups.com>
              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/
              >
              >
            • Harvey, Steven
              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
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                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]










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