Suresh Batta wrote:
> Can we now not say that, since both the ranges have comparable prime
> quantities, if a P+3 has a twin prime, then squared range
> ((p+2)^2, p^2) should have too have a twin prime?
Definitely no. The only "comparable prime quantities" are about the _expected_
number of primes based on unproven heuristics. Even if this expectation could be
proven reasonably accurate, it would not say anything about the number of twin
p/log p is the approximate number of primes below p.
The prime number theorem says it is asymptotically right. This means the
relative error (NOT the absolute error) tends to 0 when p tends to infinite.
However, as Decio notes, this and better known approximations are too poor to
say anything about the number of primes (let alone twin primes) from p to p+x
when x is much smaller than p.
It can only be used to say things like:
The _average_ number of primes in an interval of x numbers _around_ the size of
p is approximately x/log p.
Little is known about how large the deviation from such averages can be.
To illustrate possible deviations, here are the most extreme values known around
There are 11 primes (0.16 expected) among the 104-digit numbers p to p+36 for
p = 24698258*239# + 28606476153371, found by Norman Luhn and I.
There are 0 primes (30 expected) among the 93-digit numbers c to c+6378 for
c = 5629854038470321802219554908853741163682800524507382035301697914566243\
83980052820124370178769, found by Torbjörn Alm and I.
There probably exists far more extreme cases.
As far as twin primes go, there is not even anything known about averages.
Jens Kruse Andersen