- Jan 3, 2001On 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

>

> 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

>

>

>

>

>

> eGroups Sponsor

>

>

> Unsubscribe by an email to: primenumbers-unsubscribe@egroups.com

> The Prime Pages : http://www.primepages.org

>

>

>

>

>

> ---------------------------------

> Do You Yahoo!?

> Yahoo! Photos - Share your holiday photos online!

>

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

>

>

> Unsubscribe by an email to: primenumbers-unsubscribe@egroups.com

> The Prime Pages : http://www.primepages.org

Support Eric Weisstein, see http://mathworld.wolfram.com

Find the best deals on the web at AltaVista Shopping!

http://www.shopping.altavista.com - << Previous post in topic Next post in topic >>