- On Wed, 03 January 2001, Dick Boland wrote:
> Sorry Phil,

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.

>

> 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.

Phil

> -Dick

Mathematics should not have to involve martyrdom;

>

>

>

>

> 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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http://www.shopping.altavista.com - From: Milton Brown

Date: 12/21/05 16:35:16

To: Werner D. Sand; primenumbers@yahoogroups.com

Subject: RE: [PrimeNumbers] Goldbach

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.

[Non-text portions of this message have been removed]