Re: Re: Could you prove this proposition:the

• To: mathgroup at smc.vnet.net
• Subject: [mg107281] Re: [mg107269] Re: [mg107156] Could you prove this proposition:the
• From: danl at wolfram.com
• Date: Mon, 8 Feb 2010 03:33:19 -0500 (EST)
• References: <c724ed861002030412k2f8008a1x8ce30b426991a812@mail.gmail.com>

```>
> 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

Well, it's not entirely hopeless, or at least not obviously so. The "Upper
consideration (to wit, that prime(k+1)-prime(k)<k for all k) should hold
at least for all sufficiently large k.

Specifically, I think this follows from the work of Hoheisel and later
refinements such as Ingham's. In particular, the gap is shown to b
eeventually less than prime(k)^(4/5) (I use 4/5 as a value strictly
greater than 3/4). I believe the Prime Number Theorem guarantees that this
in turn is less than k for sufficiently large k; see

http://en.wikipedia.org/wiki/Prime_number_theorem

In particular see the error term bounds in comparing primepi(k) to
logintegral(k). I could be mistaken but I think they will imply the needed
inequality relating less-than-one powers of prime(k) to k.

Daniel Lichtblau
Wolfram Research

>> 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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