## [PrimeNumbers] Re: another attempt at Twin prime conjecture...

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• ... Definitely no. The only comparable prime quantities are about the _expected_ number of primes based on unproven heuristics. Even if this expectation
Message 1 of 7 , Nov 6, 2004
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
primes.

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
100 digits:

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
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