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For $n\ge6$, there are at least two primes in the interval between $n$ and $2n$. Does anyone know of an already established and accepted proof for this? A reference would be helpful.

I have read in an answer to this question - Primes between $n$ and $2n$ - that for $n\ge25$, there are at least three primes between $n$ and $2n$.

Simply checking the gaps for all $n$ up to 25 would complete the proof.

I came up with my own proof to prove something else. So I want a different proof from my own.

So my question is, is it safe to base a proof for something else on this fact?:

For $n\ge6$, there are at least two primes in the interval between $n$ and $2n$.

Where can I find a reference?

Community
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Jeffrey Young
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Ramanujan's proof of Bertrand's postulate gives this result (and more). Reference:

S. Ramanujan, A proof of Bertrand's postulate, Journal of the Indian Mathematical Society 11 (1919), pp. 181–182.

Charles
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  • The last line is $$ \pi(x) - \pi(\frac x2) \geq 1,2,3,4,5, \ldots, ::{if}:: x \geq 2, 11, 17, 29, 41, \ldots, $$ See http://ramanujan.sirinudi.org/Volumes/published/ram24.html. – lhf Apr 01 '16 at 17:18
  • That sequence of primes is called the "Ramanujan primes": see https://oeis.org/A104272 – Michael Lugo Apr 01 '16 at 17:43