The solution on Kalva seems incomplete to me... at least I see no justification why we'd only have those 3 cases, e.g. why we cannot have $ q<r$. Can anyone explain me why it is obvious?

Peter wrote: You may assume that $ p_{k}\ge\tfrac{3k}{4}$. And my question is if the statement above is always true I mean I know how to prove that $ p_{k}\geq\frac{k}{2}+3$ but that's about it ...

There are many variants on that theorem. For sufficiently high $ a$ we have a prime above $ \frac{b-1}{b}a$, I believe.

The link solutions on kalva appears not to work . Anyway I balieve that the assumtion quoted by valentin vornicu could be a wrong assution if we take into account large enough prime gaps or lagoons startin at some big enough n! +2.

Phill: Use the Wayback Machine to access Kalva (if you live in GB, use a proxy like TOR to access the Wayback Machine...). And the bounds for $ p_k$ are correct, they are a rather known generalization of the Bertrand postulate. darij

Thank you Darij