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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 . 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 and $\pi(p_{m}+m) - m > \pi(p_{n}+n) - n$.

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    What is the question?2011-07-30

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