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From: a boy on 26 May 2010 07:09
hi, Andrzej, It is easy to prove this!
please read http://en.wikipedia.org/wiki/Bertrand's_postulate
the last sentence in section #Better result shows that p[i+1]-p[i]<=i must
be TRUE when p[i] is sufficient big.

On Sun, Feb 7, 2010 at 12:11 PM, a boy <a.dozy.boy(a)gmail.com> wrote:

> 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(a)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(a)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(a)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]
>> > > >
>> > > >
>> > >
>> > >
>> >
>> >
>>
>>
>
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