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9070Re: primes, differences, a hastily constructed puzzle :-) !

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  • jbrennen
    Oct 4, 2002
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      --- In primenumbers@y..., Nathan Russell <nrussell@a...> wrote:
      > >-The prime number theorem would explain a factor of maybe a
      > > little over 1 - not 1.5, and DEFINATELY not 60
      >
      > Correction - we're talking about a high power of the natural log,
      > not the log itself. As such, the increase would be quite likely
      > over 2 but still not 60.

      I think that you are missing a big piece of the equation --
      forget about prime numbers for a minute. How many 5-row triangles
      in positive odd integers are there, where the maximum sum is 2175?
      Call that number T(5). Then, how many 6-row triangles in positive
      odd integers are there, where the maximum sum is 9069? Call that
      number T(6).

      The ratio T(6)/T(5) is at least 60, probably more. I don't have a
      clue to its exact value, but it's not too difficult to see that
      the "average" 5-row triangle with sum <= 2175 can be extended
      to a 6-row triangle with sum <= 9069 in at least 60 ways.
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