Re: -- Limit of ratio of consecutive primes = 1 ?

On May 22, 5:12 am, David C. Ullrich <dullr...@xxxxxxxxxxx> wrote:
On Wed, 21 May 2008 11:48:23 -0700 (PDT), bill <b92...@xxxxxxxxx>
wrote:



On May 21, 9:34 am, David W. Cantrell <DWCantr...@xxxxxxxxxxx> wrote:
Ray Johnstone <r...@xxxxxxxxxxxx> wrote:
On 07 May 2008 23:32:48 GMT, David W. Cantrell
<DWCantr...@xxxxxxxxxxx> wrote:

Let p(n) denote the nth prime.
It seems that, as n -> oo, limit( p(n + 1)/p(n) ) = 1.
Is that correct? If so, how can it be proven?

Does it perhaps depend on some result (unfamiliar to me) concerning how
big, for given n, the gap between p(n) and p(n + 1) can be?

So is there a limit? There seems to be no answer in this long thread.
If there is one it seems to be between 1 and 2.

The limit is 1. That was proven early in the thread.

David

The first order limit is => 1.

What in the world does that mean?

_The_ _limit_ of P(n+1)/P(n) _is_ 1.
Another way to say that would be to say that
P(n+1)/P(n) -> 1. But the _limit_ _is_ 1,
it does not "tend to 1". And nobody but
you knows what a "first order limit" is.

Since P(n+1) cannot be less that P(n) ;
the limit
must be => 1. . .
.
Twin primes quickly push the limit to 1 exactly.

Huh? That shows that limit is 1 _assuming_ two
things: (i) there are infinitely many twin primes,
(ii) the limit _exists_. But nobody knows whether
(i) is true, and (ii) is not something you're allowed
to assume here, it needs to be proved.

I understand now why you did not bother to explain this to me
earlier..
I would have to be able think like a mathematician to make any sense
out of the above
..

All of this has been based on a certain theorem known
as the Prime Number Theorem, which says that a certain
other limit is 1. That is a very deep theorem. But if you
simply assume that the limit in question exists the PNT
becomes quite simple.


Way over my head!


P(n+1) is asymptotic to (n+1) * ln (n+1) and P(n) is asymptotic to n*
ln (n)
For significantly large 'n', these two curves are virtually
identical. This
might be useful information.



Bill J

David C. Ullrich

.

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