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]<=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! > 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 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] >>> > > > >>> > > > >>> > > >>> > > >>> > >>> > >>> >>> >> >

**Re: Numerical solution of coupled nonlinear PDEs**

**Re: Working with Log**

**Weighted graphs with sum of weights determining vertex placement?**