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

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• Subject: [mg107270] 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 06:14:20 -0500 (EST)
• References: <c724ed861002030412k2f8008a1x8ce30b426991a812@mail.gmail.com> <201002041127.GAA29855@smc.vnet.net> <A725035C-2B94-425D-8644-FEE4081C4816@mimuw.edu.pl> <c724ed861002052347o335184celaa42b9629cddf85a@mail.gmail.com> <6B29BB02-5CB0-48CB-B4CE-98D8F6B18949@mimuw.edu.pl> <c724ed861002060310s6f102822k12d1037cff3ffd57@mail.gmail.com> <9A89E95D-25FF-41BB-A72F-9AAA9ED1B94F@mimuw.edu.pl> <c724ed861002062011i3f2cc58eud3d7c1614d6ffe68@mail.gmail.com> <226FDD51-C909-42B4-A055-8EE33EC031A7@mimuw.edu.pl>

```I made a mistake below writing Prime[k] where I meant PrimePi[k].

Note also, that Selberg has shown that if the Riemann hypothesis is true
then for almost all x the interval
[x,x+ f(x) Log(x)^2] contains a prime, where f(x) is any function that
tends to infinity with x. So in your case just take f(x)=
PrimePi[x]*Log[x]^2. Since by the prime number theorem PrimePi[x] is
asymptotically x/Log[x], f(x) is x*Log[x] which does tend to infinity
with x.

So it follows that, if the Riemann hypothesis is true your result is
also true for almost all n. So now you should ask someone to prove the
Riemann hypothesis. ;-)

Best wishes

Andrzej Kozlowski

On 7 Feb 2010, at 07:36, Andrzej Kozlowski wrote:

>
> On 7 Feb 2010, at 05:11, a boy wrote:
>
>> It has been proved that there exists at least a prime in the interval
(n,2n).
>
> There are of course much better results than this one (Bertrand's
postulate), although they are either asymptotic or true for some n
larger than some fixed positive integer. But they all of this type, in
other words they do not  Prime[k] in their statements. If you want to
see what results with Prime[k] look like you can see here:
>
> http://en.wikipedia.org/wiki/Prime_gap
>
> Look for Rankin's result on the lower bound for the prime gap. It's a
lower bound and is vastly more complicated than what you are proposing
(although your conjecture is almost certainly weaker than Andrica's
conjecture on the same page).
>
> Anyway: good luck.
>
> Andrzej Kozlowski
>
>
>> 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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