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k-tuples and prime packing edited

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  • Tom
    Fixed typing errors and fixed links - sorry for repost. Define kv(w) as the number of primes that can be packed into an interval of w consecutive integers.
    Message 1 of 2 , Jan 27, 2008
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      Fixed typing errors and fixed links - sorry for repost.

      Define kv(w) as the number of primes that can be packed into an
      interval of 'w' consecutive integers. This value is not a maximum
      because denser patterns are still being found in the search range up
      to w=41741, as can be seen in
      http://www.opertech.com/primes/updates.html .
      The current values for kv(w) are at
      http://www.opertech.com/primes/summary.txt , which is an ascii comma
      delimited text file.

      By the Hardy-Littlewood k-tuples conjecture, kv(w) can be defined as
      kv(w) <= pi(w+x) - pi(x) for all 'x' and 'w' when 'x' > 'w'.

      Also, a proof by Montgomery and Vaughn (1970) says,
      pi(w+x) - pi(x) < 2 * pi(w).

      After substituting, kv(w) is now
      kv(w) < 2 * pi(w),
      and removing pi(w) from both sides,
      kv(w) - pi(w) < pi(w).

      A plot of kv(w) - pi(w) is at
      http://www.opertech.com/primes/trophy.bmp .
      This plot shows the value of kv(w) - pi(w) increasing at a rate of
      about w/268-20. In fact, the slope appears to be 'increasing' or
      concave up.

      The plot provides the following
      w/268 - 20 < kv(w) - pi(w) for large w.

      Continuing, the inequality is,
      w/268 - 20 < kv(w) - pi(w) < pi(w) for large w

      and finally,
      w/268 - 20 < pi(w) for large w.

      But this inequality DOES NOT HOLD for large w, let w=e^268c then,
      e^268c / 268 - 20 < e^268c / 268c (pi(w) ~ w/ln(w))

      c*e^268c - 20*268*c < e^268c

      (c-1)*e^268c < 20*268*c

      when c=0 -1*1 < 0 -- true
      when c=1 0*1 < 20*268 -- true
      when c=2 1*e^536 < 20*536 -- false

      the inequality holds for c<=1 but at c=2

      e^536 < 20*536 is false! In fact at c=1+a, 'a' relatively small,
      the inequality fails.

      So, either the k-tuples conjecture is false, or the plot of kv(w)-pi
      (w) has an inflection point and does not have an increasing slope.
      The latter option is doubtful as kv(w)-pi(w) is a combinatorial
      object.

      Any and all input is appreciated.

      Thanx
      Tom
    • Dick
      Hi, Seems that you are establishing the magnitude of kv(w) based on the 2nd Hardy-Littlewood conjecture. The k-tuple conjecture is the first. It has already
      Message 2 of 2 , Jan 28, 2008
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        Hi,

        Seems that you are establishing the magnitude of kv(w) based on the
        2nd Hardy-Littlewood conjecture. The k-tuple conjecture is the first.
        It has already been established that the 2 conjectures are
        incompatible with each other and most believe if only one of them is
        false, it would probably be the 2nd. Assuming no errors in the rest
        of your argument, perhaps you are showing the incompatibility of the
        conjectures more so than demonstrating one of them to be false.

        http://mathworld.wolfram.com/Hardy-LittlewoodConjectures.html

        Can you clarify
        > By the Hardy-Littlewood k-tuples conjecture, kv(w) can be defined as
        > kv(w) <= pi(w+x) - pi(x) for all 'x' and 'w' when 'x' > 'w'.

        Regards,

        Dick Boland

        --- In primenumbers@yahoogroups.com, "Tom" <tom@...> wrote:
        >
        > Fixed typing errors and fixed links - sorry for repost.
        >
        > Define kv(w) as the number of primes that can be packed into an
        > interval of 'w' consecutive integers. This value is not a maximum
        > because denser patterns are still being found in the search range up
        > to w=41741, as can be seen in
        > http://www.opertech.com/primes/updates.html .
        > The current values for kv(w) are at
        > http://www.opertech.com/primes/summary.txt , which is an ascii comma
        > delimited text file.
        >
        > By the Hardy-Littlewood k-tuples conjecture, kv(w) can be defined as
        > kv(w) <= pi(w+x) - pi(x) for all 'x' and 'w' when 'x' > 'w'.
        >
        > Also, a proof by Montgomery and Vaughn (1970) says,
        > pi(w+x) - pi(x) < 2 * pi(w).
        >
        > After substituting, kv(w) is now
        > kv(w) < 2 * pi(w),
        > and removing pi(w) from both sides,
        > kv(w) - pi(w) < pi(w).
        >
        > A plot of kv(w) - pi(w) is at
        > http://www.opertech.com/primes/trophy.bmp .
        > This plot shows the value of kv(w) - pi(w) increasing at a rate of
        > about w/268-20. In fact, the slope appears to be 'increasing' or
        > concave up.
        >
        > The plot provides the following
        > w/268 - 20 < kv(w) - pi(w) for large w.
        >
        > Continuing, the inequality is,
        > w/268 - 20 < kv(w) - pi(w) < pi(w) for large w
        >
        > and finally,
        > w/268 - 20 < pi(w) for large w.
        >
        > But this inequality DOES NOT HOLD for large w, let w=e^268c then,
        > e^268c / 268 - 20 < e^268c / 268c (pi(w) ~ w/ln(w))
        >
        > c*e^268c - 20*268*c < e^268c
        >
        > (c-1)*e^268c < 20*268*c
        >
        > when c=0 -1*1 < 0 -- true
        > when c=1 0*1 < 20*268 -- true
        > when c=2 1*e^536 < 20*536 -- false
        >
        > the inequality holds for c<=1 but at c=2
        >
        > e^536 < 20*536 is false! In fact at c=1+a, 'a' relatively small,
        > the inequality fails.
        >
        > So, either the k-tuples conjecture is false, or the plot of kv(w)-pi
        > (w) has an inflection point and does not have an increasing slope.
        > The latter option is doubtful as kv(w)-pi(w) is a combinatorial
        > object.
        >
        > Any and all input is appreciated.
        >
        > Thanx
        > Tom
        >
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