- On Friday 05 November 2004 14:27, you wrote:
> <snip>

Forget about using density arguments for your `proof'. The bounds on prime

>

> Proof:

> ------

> Let p and p+2 be a twin pair. Then, p^2 and (p+2)^2 will be the their

> square range.

> If this twin exists in the primes less than p+3, then in the range of

> numbers between p^2 and (p+2)^2, the prime density should be less

> than the previous range for us to not have a twin prime in this

> square range. This would invalidate our assumption.

> The contrary will prove the conjecture.

>

> We compare these prime densities using Prime Number Theorum.

density, even assuming the Riemann Hypothesis, are much too loose for this.

Although you seem to be foregoing any kind of bounds and assuming the PNT

gives an exact answer. Sorry, but that's wrong.

> For primes including p+2, we take p+3.

Nitpick: log(p+3) != log(p) + log(3). You've repeated that mistake somewhere

>

> Primes less than p+3 can be given by p + 3 / log (p+3)

> This can be written as p + 3 / log p + log 3

> Ignoring log 3 we get, p + 3 / log p ---- 1

in the remainder of your `proof'. However this isn't what invalidates the

`proof', as you can show that log(p+k) -> log(p) as p -> oo and k is fixed;

my arguments above are the real reason why your argument fails.

Décio

[Non-text portions of this message have been removed] - Suresh Batta wrote:

> Can we now not say that, since both the ranges have comparable prime

Definitely no. The only "comparable prime quantities" are about the _expected_

> quantities, if a P+3 has a twin prime, then squared range

> ((p+2)^2, p^2) should have too have a twin prime?

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