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45Re: [PrimeNumbers] Goldbach

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  • Phil Carmody
    Jan 3, 2001
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      On Wed, 03 January 2001, Dick Boland wrote:
      > Sorry Phil,
      >
      > It was a typo. k*(k+1)/2 primes <= (p(k^2)+1)/2 and k*(k-1)/2 primes between ((p(k^2)+1)/2+1) and p(k^2). You won't find a counter example there.

      I know it was a typo, which is why in the data below I correct that typo. And with the typo corrected I do find counterexamples. Check again please.

      Phil


      > -Dick
      >
      >
      >
      >
      > Phil Carmody <fatphil@...> wrote:
      > On Wed, 03 January 2001, Dick Boland wrote:
      > > Yes I can. The distribution function is simply stated as follows,
      > >
      > > For any integer k>4, the first k^2 primes will be exactly distributed as follows:
      > >
      > > k*(k+1) primes between 1 and (p(k^2)+1)/2, and the remaining k*(k-1) primes will be distributed between ((p(k^2)+1)/2+1) and p(k^2).
      >
      > k*(k+1) + k*(k-1) == 2k^2
      >
      > So you seem to be out by a factor of 2 somewhere.
      >
      > Factoring in that factor of two...
      >
      > Table[{k,
      > k^2,
      > Prime[k^2],
      > (Prime[k^2]+1)/2,
      > PrimePi[(Prime[k^2]+1)/2],
      > k*(k+1)/2},
      > {k, 4, 8}]
      >
      > {{4, 16, 53, 27, 9, 10},
      > {5, 25, 97, 49, 15, 15},
      > {6, 36, 151, 76, 21, 21},
      > {7, 49, 227, 114, 30, 28},
      > {8, 64, 311, 156, 36, 36}}
      >
      > You seem to be saying the last two columns are the same.
      > I beg to differ.
      >
      > Let's look a bit further, at the data for k=100000-100005:
      >
      > {
      > {100000, 10000000000, 252097800623, 126048900312, 5141644677,
      > 5000050000},
      > {100001, 10000200001, 252103045511, 126051522756, 5141747035,
      > 5000150001},
      > {100002, 10000400004, 252108316073, 126054158037, 5141850524, 5000250003},
      > {100003, 10000600009, 252113577847, 126056788924, 5141953182, 5000350006},
      > {100004, 10000800016, 252118846391, 126059423196, 5142056263, 5000450010},
      > {100005, 10001000025, 252124112327, 126062056164, 5142159097, 5000550015}
      > }
      >
      > The last 2 columns really aren't that similar.
      >
      >
      > Phil
      >
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      Mathematics should not have to involve martyrdom;
      Support Eric Weisstein, see http://mathworld.wolfram.com
      Find the best deals on the web at AltaVista Shopping!
      http://www.shopping.altavista.com
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