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


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:


  • $\begingroup$ Is it an error calling the page "Choquet Theory"?? Is there an actual page on Choquet Theory that got switched with this one? $\endgroup$ Oct 17 '16 at 12:38

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.

  • 5
    $\begingroup$ This result (which Erdős indeed proved), was originally proven by Breusch, according to this and other sources: books.google.com/… $\endgroup$
    – Steve Kass
    Oct 5 '15 at 0:50
  • $\begingroup$ @SteveKass It's also written on Breusch's Wikipedia page with $3$ references. $\endgroup$
    – user236182
    Oct 5 '15 at 0:56
  • 3
    $\begingroup$ It seems the MathWorld link might be misleading. It's not clear if there's a proof of this fact in the reference. Also, is it related to Choquet theory? And the MathWorld page should contain more references (any of these given here). Someone should edit the MathWorld link. $\endgroup$
    – user236182
    Oct 5 '15 at 1:06
  • $\begingroup$ The full text of Breusch’s paper is available here. eudml.org/doc/168326 It doesn’t address your original question, but it might be helpful to some readers. $\endgroup$
    – Steve Kass
    Oct 5 '15 at 2:02

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