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

• To: mathgroup at smc.vnet.net
• Subject: [mg107156] 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: Thu, 4 Feb 2010 06:27:07 -0500 (EST)
• References: <c724ed861002030412k2f8008a1x8ce30b426991a812@mail.gmail.com>

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]

• Follow-Ups:
  • Re: Could you prove this proposition:the i-th prime gap p[i+1]-p[i]<=i
    • From: Andrzej Kozlowski <akoz@mimuw.edu.pl>