Re: Could you prove this proposition:the i-th prime gap
- To: mathgroup at smc.vnet.net
- Subject: [mg107265] 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 06:13:19 -0500 (EST)
- References: <c724ed861002030412k2f8008a1x8ce30b426991a812@mail.gmail.com>
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] > > > > > > > > > > > >