Re: Could you prove this proposition:the i-th prime gap

*To*: mathgroup at smc.vnet.net*Subject*: [mg107268] Re: [mg107156] Could you prove this proposition:the i-th prime gap*From*: a boy <a.dozy.boy at gmail.com>*Date*: Sun, 7 Feb 2010 06:13:56 -0500 (EST)*References*: <c724ed861002030412k2f8008a1x8ce30b426991a812@mail.gmail.com>

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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**Re: Could you prove this proposition:the i-th prime gap**

**A New Scientist article verified with Mathematica**