Re: Could you prove this proposition:the i-th prime gap

• To: mathgroup at smc.vnet.net
• Subject: [mg107268] Re: [mg107156] Could you prove this proposition:the i-th prime gap
• From: a boy <a.dozy.boy at gmail.com>
• Date: Sun, 7 Feb 2010 06:13:56 -0500 (EST)
• References: <c724ed861002030412k2f8008a1x8ce30b426991a812@mail.gmail.com>

```It has been proved that there exists at least a prime in the interval
(n,2n).
p[i+1]-p[i]<=i  iff there exists at least a prime in the interval
(n,n+Pi(n)]
This is an improvement for the upper bound of prime gap, so I think it is
not very difficult.
For the simpleness and elegance of the form p[i+1]-p[i]<=i, I think someone
can prove this. We should be more optimistic!

On Sat, Feb 6, 2010 at 10:11 PM, Andrzej Kozlowski <akoz at mimuw.edu.pl>wrote:

> > 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[1] = (p[n]-p[n-1]) + (p[n-1]-p[n-2]) + ... + (p[2]-p[1]) <=
> (n-1)+ (n-2) + ... + 1 == (n-1) n/2
> > >
> > > hence
> > >
> > > p[n]<= p[1]+ (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]+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]
> > > >
> > > >
> > >
> > >
> >
> >
>
>

```

• Prev by Date: Re: Re: What does & mean?
• Next by Date: Re: Re: What does & mean?
• Previous by thread: Re: Could you prove this proposition:the i-th prime gap
• Next by thread: A New Scientist article verified with Mathematica