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It's easy to find Ramanujan's proof of Ramanujan primes:

Ramanujan's Proof

Wikipedia mentions that Paul Erdős also had a proof:

Wikipedia article on Bertrand's Postulate

Does anyone know the full citation for Erdős's proof that for any number $n$, there exists a prime $p$ such that for all $x \ge p$, there are $n$ primes between $x$ and $2x$?

Thanks very much,


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I think this is a reasonable question (as reference request). That said, does the Erdos paper cited in the references of that Wikipedia article do what you want? –  Yemon Choi Aug 22 '11 at 4:45
The reference in Wikipedia (1934) does the full result with $n$ primes. The original paper on the Bertrand Postulate itself was in Acta. Litt. Ac. Sci. (Szeged) vol 5 (1932) pp 194-198. The 1932 article was responsible for: $$ $$ Chebyshev said it, and I say it again, $$ $$ There is always a prime between $n$ and $2n$ –  Will Jagy Aug 22 '11 at 5:08
You may also want to check "Proofs from the book", where Erdős's original argument is presented (with a picture of the paper), together with a further simplification he suggested (with a picture of the note). –  Andres Caicedo Aug 22 '11 at 5:22
Will, thanks very much! I found it in German: math-inst.hu/~p_erdos/1932-01.pdf And Google's translation to English: translate.google.com/… –  Larry Freeman Aug 22 '11 at 5:35
Yemon, got it. Sorry that I didn't understand. I should have mentioned that when I checked the Erdos paper on Schur and Sylvester, I did not find the details that I expected. Here's the link I used to the Erdos paper that's cited in the Wikipedia article: profs.sci.univr.it/~bellin/philsci/Erdos.pdf –  Larry Freeman Aug 22 '11 at 9:46

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