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• Sorry Phil, It was a typo. k*(k+1)/2 primes
Message 1 of 45 , Jan 3, 2001
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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.

-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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• From: Milton Brown Date: 12/21/05 16:35:16 To: Werner D. Sand; primenumbers@yahoogroups.com Subject: RE: [PrimeNumbers] Goldbach These messages about
Message 45 of 45 , Dec 21 1:46 PM
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From: Milton Brown
Date: 12/21/05 16:35:16

These messages about Goldbach's Conjecture are not
supposed to be to this mailing list (also the Riemann Hypothesis).

There are separate mailing lists for these.

Kermit says.

Milton! You surprise me.

Goldbach's conjecture IS about prime numbers. It's doesn't matter that
there exist mailing lists specifically about Goldbach's conjecture.

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