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k-tuples and prime packing

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  • Tom
    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
    Message 1 of 1 , Jan 26, 2008
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      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 www.opertech.com/primes/updates.html .
      The current values for kv(w) are at
      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 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 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

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

      e^268*2 < 20*268*2 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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