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

• To: mathgroup at smc.vnet.net
• Subject: [mg107265] 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 06:13:19 -0500 (EST)
• References: <c724ed861002030412k2f8008a1x8ce30b426991a812@mail.gmail.com>

```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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