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

• To: mathgroup at smc.vnet.net
• Subject: [mg107236] 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: Sat, 6 Feb 2010 03:24:48 -0500 (EST)
• References: <c724ed861002030412k2f8008a1x8ce30b426991a812@mail.gmail.com> <201002041127.GAA29855@smc.vnet.net>

```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]
>
>

```