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Re: (p[n+2]-p[n+1])(p[n+1]-p[n])/4

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  • Zak Seidov
    ... All of that is lost (unless, Zak, you want to cut and paste and post ... Also I did not receive this! ... message, ... a ... post ... is ... the
    Message 1 of 22 , Oct 1, 2003
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      >
      All of that is lost (unless, Zak, you want to cut and paste and post
      > for me).

      Also I did not receive this!

      --- In primenumbers@yahoogroups.com, "Adam" <a_math_guy@y...> wrote:
      > Shoot, last Friday I sent this fairly long response to this
      message,
      > which isn't posted here, because this is the only yahoo group I am
      a
      > part of for which "reply" defaults to the poster and not the whole
      > group.
      >
      > Sigh.
      >
      > All of that is lost (unless, Zak, you want to cut and paste and
      post
      > for me).
      >
      > Anyway, in that intended post I reasoned that the +1 mod 3 residue
      is
      > impossible, and that the other residues should happen, if searched
      > deep enough. As a proof of concept I list the following p values
      > that have [p(n+2)-p(n+1)]*[p(n+1)-p(n)]/4=194:
      >
      > p(n) from {1374538987513, 1374682325293, 1374446799889,
      > 1374488450563, 1374402345229, 1374600127909, 1374415063363,
      > 1374735847123, 1374546494143, 1374429446293, 1374472611199,
      > 1374810758323, 1374520532653, 1374801030553, 1374811265683,
      > 1374679474459, 1374724625689, 1374637137469, 1374748849219,
      > 1374445952569, 1374505434703, 1374691452103, 1374702725089,
      > 1374770337163, 1374598127599, 1374757434889, 1374584932489,
      > 1374696888559, 1374406298959, 1374724738573, 1374429655633,
      > 1374721538119, 1374775427743, 1374475007509, 1374593340163,
      > 1374523688173, 1374670818133, 1374665432893, 1374734529673,
      > 1374786320833, 1374515455399}
      >
      > I believe that many, if not all, of the other -1 mod 3 numbers on
      the
      > list would also 'fall' to a search of sufficient depth.
      >
      > Adam
      >
      > --- In primenumbers@yahoogroups.com, "Zak Seidov" <seidovzf@y...>
      > wrote:
      > > Just sent to OEIS:
      > >
      > > 4,7,10,13,16,19,22,25,28,31,34,37,40,
      > > 43,46,49,52,55,58,61,64,67,70,73,76,
      > > 79,82,83,85,88,89,91,94,97,100,101,
      > > 103,106,107,109,112,113,115,118,121,
      > > 124,127,130,131,133,136,137,139,142,
      > > 145,148,149,151,154,157,158,160,163,
      > > 166,167,169,172,173,175,178,179,181,
      > > 184,187,190,191,193,194,196,197,199
      > >
      > > Numbers which can not be of form
      > > (p[n+2]-p[n+1])(p[n+1]-p[n])/4.
      > >
      > > Some cases are proved easily,
      > > some are only guesses checked
      > > for n<=2,000,000.
      > >
      > > SHANA TOVA!!
      > >
      > > Zak
    • Zak Seidov
      Shalom R. A. Twain, 1) OEIS assumes scientific level or smth like, according which you may reject most of seqs in it... 2) guesses in any sci sense are
      Message 2 of 22 , Oct 1, 2003
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        Shalom R. A. Twain,

        1) OEIS assumes "scientific level" or smth like,
        according which you may reject most of seqs in it...
        2) "guesses" in any sci sense are more important
        than "established facts"
        3) my this particular seq was labeled as "bad sequence"
        (see copy of Neil's message -
        sorry if someone reads it twise or trice)
        and I guess that it will be removed from OEIS
        4)in my (weakest) exuse -
        i am only amateur and fan
        5)"knowledge causes only sadness"
        may i only add: "extra" knowledge
        6)My deepest respects to all NT gurus
        7)Zak

        %%%%%%%%%%%%%%%%%%%%end of copy of Neil's message%%%%%%%%%%%%
        From: N. J. A. Sloane [njas@...]
        Sent: 1:34 02/10/03
        Subject: bad sequence
        this just came in from a correspondent:
        ID Number: A087656
        URL: http://www.research.att.com/projects/OEIS?Anum=A087656
        Sequence: 4,7,10,13,16,19,22,25,28,31,34,37,40,43,46,49,52,55,58,61,
        <skip>
        Name: Numbers that can not be of form (p[n+2]-p[n+1])(p[n+1]-p
        [n])/4, where p[i] is i-th prime.
        Comments: Some cases are proved easily, some are only guesses checked
        only up n<=2000000.
        See also: Adjacent sequences: A087653 A087654 A087655 this_sequence
        A087657 A087658 A087659
        Keywords: nonn,new
        Offset: 4
        Author(s): Zak Seidov (seidovzf(AT)yahoo.com), Sep 26 2003
        >>>

        The sequence is bogus because:

        1) All numbers of the form 3n+1 (>1) are in it
        2) Subject to heuristically safe conjecture, no number not of the form
        3n+1 can be in it.

        In more detail -

        (1) is true as 12n+4 factors into either(3a+1)(3b+1) or (3a+2)(3b+2),
        and
        neither of those can be the fingerprint for an admissible 3-tuple
        apart
        from {3,5,7} -> (2)*(2)/4 = 1.

        (2) is probably true because any {0,2,6n+2} is an admissible 3-tuple,
        and
        using the same heuristics as are behind the k-tuple conjecture, but
        based
        on the principles behind Dirichlet's Theorem. An arithmetic
        progression
        a+i.b can be created with known composites at a+i.p+1, and a+i.p+
        {3..6n+1}
        using the chinese remainder theorem. This AP should have a density of
        prime k-tuples in proportion (by a fixed constant, realted to the
        multiplier b) with the number of tuplets that arbitrary integers would
        yield - which by Hardy & Littlewood's (2nd) conjecture is infinite.

        e.g. the first number not of the form 3n+1 in the list, 83 fails
        because

        (00:34) gp > p=257987875972449177033341526073139
        257987875972449177033341526073139
        (00:35) gp > isprime(p)
        1
        (00:35) gp > q=nextprime(p+1)
        257987875972449177033341526073141
        (00:35) gp > r=nextprime(q+1)
        257987875972449177033341526073307
        (00:35) gp > (r-q)*(q-p)/4
        83

        (not the smallest, I made a typo in my script, but a number jumped out
        within seconds anyway!)
        %%%%%%%%%%%%%%%%%%%%end of copy of Neil's message%%%%%%%%%%%%

        --- In primenumbers@yahoogroups.com, "ratwain" <ratwain@y...> wrote:
        > --- In primenumbers@yahoogroups.com, "Zak Seidov" <seidovzf@y...>
        wrote:
        > > Just sent to OEIS:
        >
        > I hope you haven't. I don't really think the Online Encyclopedia
        of Integer Sequences was created for "guesses" -- your sequence
        doesn't belong there.
        >
        > > 4,7,10,13,16,19,22,25,28,31,34,37,40,
        <skip>
        > >
        > > Numbers which can not be of form
        > > (p[n+2]-p[n+1])(p[n+1]-p[n])/4.
        >
        > All numbers of the form 3n+1 are part of the set, and the rest
        aren't. I'm sure there are people on the list who can give you some
        more hints if you need them, so I'll just say "k-tuple conjecture."
        >
        > R. A. Twain
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