       Re: Could you prove this proposition:the i-th prime gap p[i+1]-p[i]<=i

• To: mathgroup at smc.vnet.net
• Subject: [mg107272] Re: [mg107156] Could you prove this proposition:the i-th prime gap p[i+1]-p[i]<=i
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Sun, 7 Feb 2010 06:14:44 -0500 (EST)
• References: <c724ed861002030412k2f8008a1x8ce30b426991a812@mail.gmail.com> <201002041127.GAA29855@smc.vnet.net> <A725035C-2B94-425D-8644-FEE4081C4816@mimuw.edu.pl> <c724ed861002052347o335184celaa42b9629cddf85a@mail.gmail.com> <6B29BB02-5CB0-48CB-B4CE-98D8F6B18949@mimuw.edu.pl> <c724ed861002060310s6f102822k12d1037cff3ffd57@mail.gmail.com>

```> I think it is not difficult to prove the proposition,but I can't do
this still.

You think or you hope? I think it is going to be extremely difficult to
prove it and the reason is that nothing of this kind has been proved
even though other people also have computers and eyes. There are some
very weak asymptotic results and there are conjectures, for which the
only evidence comes from numerical searches. The best known is Andrica's
conjecture which states that  Sqrt[Prime[i+1]]-Sqrt[Prime[i]]<1 and
appears to be stronger than yours, but nobody has any idea how to prove
that. In fact, nobody can prove that
Limit[Sqrt[Prime[n+1]]-Sqrt[Prime[n]],n->Infinity]=0 (this has been
open since 1976), and in fact there is hardly any proved statement of
this kind. So what is the reason for your optimism?

Andrzej Kozlowski

On 6 Feb 2010, at 12:10, a boy wrote:

> Yes,I want the proof of the fact that p[i+1]-p[i]<=i.
> I think it is not difficult to prove the proposition,but I can't do
this still.
> If he or she give me a proof , I will be very happy and appreciate him
or her!
>
> On Sat, Feb 6, 2010 at 6:50 PM, Andrzej Kozlowski <akoz at mimuw.edu.pl>
wrote:
> Oh, I see. You meant you want the proof of the fact that
p[i+1]-p[i]<=i? I misunderstood your question I thought you wanted to
see the trivial deduction of the statement you had below that.
>
> But, considering that practically nothing is known about upper bounds
on prime number gaps p[i+1]-p[i] in terms of i (all known results
involve bounds in terms of p[i] and these are only asymptotic), this
kind of proof would be a pretty big result so, in the unlikely event any
of us could prove it, would you except him or her just to casually post
it here?  ;-)
>
> Andrzej Kozlowski
>
>
>
> On 6 Feb 2010, at 08:47, a boy wrote:
>
> > When I was observing the prime gaps, I conjectured
> > p[i+1]-p[i]<=i
> >
> > This means there is at least a prime between the interval =
(n,n+Pi(n)].  I verified this by Mathematica and searched in web, but I
can't prove this yet.
> >
> > On Sat, Feb 6, 2010 at 4:17 AM, Andrzej Kozlowski =
<akoz at mimuw.edu.pl> wrote:
> > Hmm... this is a little weird - how come you know this if you can't
prove it? This is one of those cases where knowing something is
essentially the same as proving it... but anyway:
> >
> > p[n]-p = (p[n]-p[n-1]) + (p[n-1]-p[n-2]) + ... + (p-p) <==
(n-1)+ (n-2) + ... + 1 == (n-1) n/2
> >
> > hence
> >
> > p[n]<= p+ (n-1)n/2 = 2 + (n-1)n/2
> >
> > Andrzej Kozlowski
> >
> >
> > On 4 Feb 2010, at 12:27, a boy wrote:
> >
> > > Hello!
> > > By my observation, I draw a conclusion: the i-th prime gap
> > > p[i+1]-p[i]<=i
> > > Could you give me a simple proof for the proposition?
> > >
> > > p[i+1]-p[i]<=i  ==>  p[n]<p+1+2+..+ n-1=2+n(n-1)/2
> > >
> > > Mathematica code:
> > > n = 1;
> > > While[Prime[n + 1] - Prime[n] <= n, n++]
> > > n
> > >
> > > Clear[i];
> > > FindInstance[Prime[i + 1] - Prime[i] > i && 0 < i, {i}, Integers]
> > >
> > >
> >
> >
>
>

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