open conjecture (related to Bertrand postulate)

heartwork

The conjecture states:
Always exists a prime number between two consecutive squares.
We try to count all the composite numbers between $n^2$ and $(n+1)^2$ and if we succed to proof that are less than $2n$ this is done.
First count composite numbers multiplier of 2, then numbers multiplier of 3, but not of 2, then multipliers of 5, but not of 2 and 3, and so on, until $p_{k}<=n$ .
Between that $2n$ numbers , if $k$ divides $2n$ , exactly $2n/k$ are multipliers of $k$ .
If not, we might have $[2n/k]$ or $[2n/k]+1$ multipliers of $k$ , so with some error (less than 1) we should consider the same number of multipliers $2n/k$ - not integer.
If we can succesfully manage that total error $E_{k}$ the conjecture might be proved if:
For any such prime $p_k<=n$ we have:
$s_{k}+E_{k}/2n<1$ , where $s_{k}$ is:
$1/2 + 1/3(1-1/2) + 1/5(1-1/2)(1-1/3) + 1/7(1-1/2)(1-1/3)(1-1/5) + ... + 1/{p_k}(1-1/2)(1-1/3)...(1-1/{p_{k-1}})$ ,
and $s_{k}=1-1/(1-1/2)(1-1/3)...(1-1/{p_{k-1}})(1-1/{p_{k}})$
obviously less than 1. (tends to 1 as $p_k$ grows unbounded)
Then the huge conjecture is done Mr. Green

Do you want to give a try?

**Posted:** Tue Aug 10, 2004 3:46 am

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heartwork

As far as I know the best related result already proved is that always between $n$ and $n+O(n^{6/11})$ there is a prime in 1992. Does someone know more about this? Did you see a proof for this result somewhere on-line? (if yes, please supply a link here)

**Posted:** Thu Aug 12, 2004 4:19 am

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heartwork

No ideas?

**Posted:** Wed Aug 18, 2004 2:47 am

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Gyan

But an easy problem is to prove that if p and q are two consucative odd primes, prove that (p+q)/2 is composit.

**Posted:** Wed Aug 18, 2004 1:11 pm

bugzpodder · Yang-Mills Theory Offline Joined: 12 Sep 2003 Posts: 659

since p,q is odd then (p+q)/2 is an integer, and since it is between p and q, two consecutive primes, of course its composite

**Posted:** Wed Aug 18, 2004 2:13 pm

heartwork

It is known that always exists a prime number between $n$ and $n+O(n^{6/11})$ (1992).
This old conjecture states that:
always exists a prime number between $n$ and $n+O(n^{1/2})$ .
(in a more general form)
I wonder if the following:
$p_{k} < {e^{1/((1-1/3)(1-1/5)...(1-1/p_{k}))}$
is true?
(if that $E_{k}<=O(k)$ then the 1st inequality is true if this one is true, but in the opposite sense! - this leads to conclusion that evaluation $E_{k}=O(k)$ is weak (even i am not able to prove it - 1st brute approach leads to $E_{k}<2^k$ !))

**Posted:** Thu Aug 19, 2004 1:34 am

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riemann · New Member Offline Joined: 11 Apr 2005 Posts: 3

can you give me some aoolications of bertrand's postulate

**Posted:** Mon Apr 11, 2005 9:08 am

riemann · New Member Offline Joined: 11 Apr 2005 Posts: 3

can you give me some applications of bertrand's postulate

**Posted:** Mon Apr 11, 2005 9:10 am

DCo

Applications of bertrand? No factorial is a perfect square.
f you can find 2n consecutive numbers so that [2n/k]+1 of them are divisible by k, I'll eat my hat.

**Posted:** Sat Apr 23, 2005 5:20 pm

Phelpedo

What if k is 1? Then you have to eat your hat!

**Posted:** Tue Apr 26, 2005 2:54 pm

DCo

Er... if I have 2n integers, are 2n+1 of them divisible by 1??

**Posted:** Tue Apr 26, 2005 6:54 pm

riemann · New Member Offline Joined: 11 Apr 2005 Posts: 3

thanks for the help I ask you if we work on an elementary proof of the riemann hypothesis using the galaria 's and robin's results

**Posted:** Sun May 01, 2005 6:26 am

heartwork

**Posted:** Wed May 18, 2005 1:15 am

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