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

• To: mathgroup at smc.vnet.net
• Subject: [mg107251] Re: [mg107156] Could you prove this proposition:the i-th prime gap
• From: a boy <a.dozy.boy at gmail.com>
• Date: Sat, 6 Feb 2010 03:27:31 -0500 (EST)
• References: <c724ed861002030412k2f8008a1x8ce30b426991a812@mail.gmail.com>

```When I was observing the prime gaps, I conjectured
p[i+1]-p[i]<=iThis 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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