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It is well known that primes of form $4k+3$, call them $3=q_1 < q_2 < \dots$ satisfy $q_{n+1}/q_n\rightarrow 1$ (and even $q_n=\frac{n}{2\log n}(1+o(1))$). I would be glad to see results of Dusart-type (Strong Bertrand postulate) with concrete bounds for $N$ such that $q_{n+1} < 1.1 q_n$ provided $q_n > N$ or something like this (I am not sure that I need a multiple exactly $1.1$)

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I would suggest contacting Pierre Dusart directly. Whether he has any results in that direction or not, I'm sure he'd know what the state of the art is. – Charles Feb 18 '11 at 18:54

Have a look at (Satz 9 and the preceeding calculations):

1935-10 P. Erdõs: Über die Primzahlen gewisser arithmetischer Reihen (in German), Math. Z. 39 (1935), 473--491; Zentralblatt 10,293.

Available at:

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Thank you! Though this seem to work for intervals $[n,2n]$ and I do not see how to change arguments for $[n,cn]$ with $c$ sufficiently close to 1. – Fedor Petrov Feb 14 '11 at 8:50
It is not known how to make these kinds of elementary arguments work with $c$ arbitrarily close to $1$. I don't know how to get it better than $c=2$ but it might be possible. – Felipe Voloch Feb 14 '11 at 8:59
I would be happy with non-elementary methods (with $L$-functions probably) aswell, if they give results of such type. – Fedor Petrov Feb 14 '11 at 16:36

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