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$p_i$ is the $i^{\rm th}$ prime. $\pi(x)$ is prime counting function.

Firstly, I think that Prime gap inequality holds true for any $i>0$: $p_{i+1} - p_{i} \leq i$.

Very often, $\pi(p_{m}+m) - m \leq \pi(p_{n}+n) - n$ if $m<n$ . However, there exist counter examples. $\pi(17+7)-7 > \pi(19+8)-8$ . I conjecture there exists infinite this sort of counter examples. In math words,

Conjecture:$ \forall x , \exists m,n$, satisfy $ x<m<n $ and $\pi(p_{m}+m) - m > \pi(p_{n}+n) - n$.

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I have editing the typesetting a bit. Most significantly, I have replaced the $p[i]$ notation with the more standard subscript notation $p_i$. I haven't changed the wording otherwise. Hope this is ok. (My edit will be visible to all only after it is peer-reviewed.) –  Srivatsan Jul 30 '11 at 6:02
    
very kind of you! –  a boy Jul 30 '11 at 6:17
    
How much and what kind of evidence do you have supporting your conjecture? Did you run some computer experiments (this should be rather easy to do)? –  t.b. Jul 30 '11 at 6:55
    
@Theo p = Prime; pi = PrimePi; Table[pi[p[n] + n] - n, {n, 400, 500}] –  a boy Jul 30 '11 at 7:37
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Thanks. I don't have Mathematica, so I can't use your code (well - I could achieve that easily with sage, too, of course, if I wanted to). However, I was intending to suggest something that doesn't take a few microseconds only but would be a serious test. I mean, $\sim 35'000$ isn't very big. –  t.b. Jul 30 '11 at 7:55

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