- Fixed typing errors and fixed links - sorry for repost.

Define kv(w) as the number of primes that can be packed into an

interval of 'w' consecutive integers. This value is not a maximum

because denser patterns are still being found in the search range up

to w=41741, as can be seen in

http://www.opertech.com/primes/updates.html .

The current values for kv(w) are at

http://www.opertech.com/primes/summary.txt , which is an ascii comma

delimited text file.

By the Hardy-Littlewood k-tuples conjecture, kv(w) can be defined as

kv(w) <= pi(w+x) - pi(x) for all 'x' and 'w' when 'x' > 'w'.

Also, a proof by Montgomery and Vaughn (1970) says,

pi(w+x) - pi(x) < 2 * pi(w).

After substituting, kv(w) is now

kv(w) < 2 * pi(w),

and removing pi(w) from both sides,

kv(w) - pi(w) < pi(w).

A plot of kv(w) - pi(w) is at

http://www.opertech.com/primes/trophy.bmp .

This plot shows the value of kv(w) - pi(w) increasing at a rate of

about w/268-20. In fact, the slope appears to be 'increasing' or

concave up.

The plot provides the following

w/268 - 20 < kv(w) - pi(w) for large w.

Continuing, the inequality is,

w/268 - 20 < kv(w) - pi(w) < pi(w) for large w

and finally,

w/268 - 20 < pi(w) for large w.

But this inequality DOES NOT HOLD for large w, let w=e^268c then,

e^268c / 268 - 20 < e^268c / 268c (pi(w) ~ w/ln(w))

c*e^268c - 20*268*c < e^268c

(c-1)*e^268c < 20*268*c

when c=0 -1*1 < 0 -- true

when c=1 0*1 < 20*268 -- true

when c=2 1*e^536 < 20*536 -- false

the inequality holds for c<=1 but at c=2

e^536 < 20*536 is false! In fact at c=1+a, 'a' relatively small,

the inequality fails.

So, either the k-tuples conjecture is false, or the plot of kv(w)-pi

(w) has an inflection point and does not have an increasing slope.

The latter option is doubtful as kv(w)-pi(w) is a combinatorial

object.

Any and all input is appreciated.

Thanx

Tom - Hi,

Seems that you are establishing the magnitude of kv(w) based on the

2nd Hardy-Littlewood conjecture. The k-tuple conjecture is the first.

It has already been established that the 2 conjectures are

incompatible with each other and most believe if only one of them is

false, it would probably be the 2nd. Assuming no errors in the rest

of your argument, perhaps you are showing the incompatibility of the

conjectures more so than demonstrating one of them to be false.

http://mathworld.wolfram.com/Hardy-LittlewoodConjectures.html

Can you clarify> By the Hardy-Littlewood k-tuples conjecture, kv(w) can be defined as

Regards,

> kv(w) <= pi(w+x) - pi(x) for all 'x' and 'w' when 'x' > 'w'.

Dick Boland

--- In primenumbers@yahoogroups.com, "Tom" <tom@...> wrote:

>

> Fixed typing errors and fixed links - sorry for repost.

>

> Define kv(w) as the number of primes that can be packed into an

> interval of 'w' consecutive integers. This value is not a maximum

> because denser patterns are still being found in the search range up

> to w=41741, as can be seen in

> http://www.opertech.com/primes/updates.html .

> The current values for kv(w) are at

> http://www.opertech.com/primes/summary.txt , which is an ascii comma

> delimited text file.

>

> By the Hardy-Littlewood k-tuples conjecture, kv(w) can be defined as

> kv(w) <= pi(w+x) - pi(x) for all 'x' and 'w' when 'x' > 'w'.

>

> Also, a proof by Montgomery and Vaughn (1970) says,

> pi(w+x) - pi(x) < 2 * pi(w).

>

> After substituting, kv(w) is now

> kv(w) < 2 * pi(w),

> and removing pi(w) from both sides,

> kv(w) - pi(w) < pi(w).

>

> A plot of kv(w) - pi(w) is at

> http://www.opertech.com/primes/trophy.bmp .

> This plot shows the value of kv(w) - pi(w) increasing at a rate of

> about w/268-20. In fact, the slope appears to be 'increasing' or

> concave up.

>

> The plot provides the following

> w/268 - 20 < kv(w) - pi(w) for large w.

>

> Continuing, the inequality is,

> w/268 - 20 < kv(w) - pi(w) < pi(w) for large w

>

> and finally,

> w/268 - 20 < pi(w) for large w.

>

> But this inequality DOES NOT HOLD for large w, let w=e^268c then,

> e^268c / 268 - 20 < e^268c / 268c (pi(w) ~ w/ln(w))

>

> c*e^268c - 20*268*c < e^268c

>

> (c-1)*e^268c < 20*268*c

>

> when c=0 -1*1 < 0 -- true

> when c=1 0*1 < 20*268 -- true

> when c=2 1*e^536 < 20*536 -- false

>

> the inequality holds for c<=1 but at c=2

>

> e^536 < 20*536 is false! In fact at c=1+a, 'a' relatively small,

> the inequality fails.

>

> So, either the k-tuples conjecture is false, or the plot of kv(w)-pi

> (w) has an inflection point and does not have an increasing slope.

> The latter option is doubtful as kv(w)-pi(w) is a combinatorial

> object.

>

> Any and all input is appreciated.

>

> Thanx

> Tom

>