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

• To: mathgroup at smc.vnet.net
• Subject: [mg107254] 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 06:11:12 -0500 (EST)
• References: <c724ed861002030412k2f8008a1x8ce30b426991a812@mail.gmail.com> <201002041127.GAA29855@smc.vnet.net> <A725035C-2B94-425D-8644-FEE4081C4816@mimuw.edu.pl> <c724ed861002052347o335184celaa42b9629cddf85a@mail.gmail.com>

```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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