k-tuples and prime packing
- 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 www.opertech.com/primes/updates.html .
The current values for kv(w) are at
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 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
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
w/268 - 20 < pi(w) for large w.
But this inequality DOES 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
the inequality hold for c<=1 but at c=2
e^268*2 < 20*268*2 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
Any and all input is appreciated.