<http://mathworld.wolfram.com/ChoquetTheory.html>


Is the claim in the link true? Here's the reference given there:


[https://www.renyi.hu/~p\_erdos/1934-01.pdf](https://www.renyi.hu/%7Ep_erdos/1934-01.pdf)



> 
> Erdős proved that there exist at least one prime $\equiv 1\pmod{4}$ and at least one prime $\equiv 3\pmod{4}$ between $n$ and $2n$ for all $n>6$.
> 
> 
> References:
> *Erdős, P. "A Theorem of Sylvester and Schur." J. London Math. Soc. 9, 282-288, 1934.*
> 
> 
> 


Also: <https://en.wikipedia.org/wiki/Talk:Bertrand%27s_postulate#Dubious_statement>


My question:



> 
> Can you find the proof of this in the given pdf (maybe somewhere else)?
> 
> 
> 


EDIT: For those interested, there's a generalization of this fact:


<https://books.google.lt/books?id=tmORL-UYOyEC&pg=PA386&lpg=PA386&redir_esc=y#v=onepage&q&f=false>



Did Erdős prove there are two primes $4a+1, 4b+3$ between between $n$ and $2n$?
Yes, see the final page of 


P. Erdos: Bizonyos számtani sorok törzsszámairól (On primes in some arithmetic progressions, in Hungarian), Bölcsészdoktori értekezés , Sárospatak, 1934, 1--20. 


or its German translation:


P. Erdos: Über die Primzahlen gewisser arithmetischer Reihen (in German), Math. Z. 39 (1935), 473--491.


It looks like Erdos's proof only gives the result *directly* when $n > 6000$. But as he remarks, it is easy to lower this limit by a short, direct computation.


