Distribution of twin primes

[Pfoertner, Hugo]

David Wilson wrote:
> Hugo Pfoertner wrote: 
>> The interesting thing is, that I couldn't find more terms in the latter
>> sequence up to n=46340 (n^2~=32bit signed integer 2.147*10^9). This
suggests
>> the rather bold conjecture, that all intervals beyond [123^2,124^2]
contain
>> at least one twin prime pair. Proving seems impossible. Anybody
interested
>> in extending the checked range? 
> Maybe not impossible, but whereas this conjecture implies the Twin Primes 
> conjecture, it will presumably be difficult.

In the meantime I found "Increasing gaps among twin primes: size."
http://www.research.att.com/projects/OEIS?Anum=A036063
after putting the record gaps found by my own program
2 6 12 18 30 36 72 150 168 210 282 372 498 ....
between successive twin prime pairs into
superseeker. (offset 2 and first two terms missing).

I plotted the record gaps and the location of their first occurrence
http://www.research.att.com/projects/OEIS?Anum=A036061
into a diagram with x-axis = sqrt(n) (sorry axis titel "n") together
with the length of the intervals between successive squares.

The reason why no more empty intervals without twins are found beyond a
certain (low) limit becomes visible in
http://www.randomwalk.de/sequences/twingap.pdf
Beyond sqrt(n)=1000 (x-Axis) even the record gap curve drops significantly
below the successive square interval length curve and the distance seems
to increase for larger n.
The average number of twins per interval=(x+1)^2-x^2=2*x+1 (lower curve) is
far
below (using the formula pi_2 = C * PI_2 * x / (ln(x))^2 with PI_2=twin
prime
constant ~= 0.66.. and taking Haugland's C=6.8325.
See http://mathworld.wolfram.com/TwinPrimes.html

Hugo