## k-tuples and prime packing edited

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• 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.
Message 1 of 2 , Jan 27, 2008
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
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
Message 2 of 2 , Jan 28, 2008
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
> kv(w) <= pi(w+x) - pi(x) for all 'x' and 'w' when 'x' > 'w'.

Regards,

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
> 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
>
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