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

• To: mathgroup at smc.vnet.net
• Subject: [mg112116] Re: Could you prove this proposition:the i-th prime gap p[i+1]-p[i]<=i
• From: a boy <a.dozy.boy at gmail.com>
• Date: Tue, 31 Aug 2010 04:16:49 -0400 (EDT)

```Now, I can prove that p[i+1]-p[i]<=i. please visit
http://my.unix-center.net/~martian/math%20album/%E9%A3%9E%E7%BF%94%E7%9A%84%E6%B5%B7%E6%B2%99%20~%20a%20boy's%20math%20album%20-%201/%E6%B5%81%E6%B0%B4%E9%A3%9E%E9%B8%9F%20-%20p[i+1]-p[i]&lt=i.nb

On Wed, May 26, 2010 at 4:38 PM, a boy <a.dozy.boy at gmail.com> wrote:

> hi, Andrzej, It is easy to prove this!
> 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 at 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 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]
>>> > > >
>>> > > >
>>> > >
>>> > >
>>> >
>>> >
>>>
>>>
>>
>

```

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