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It is known that there exists a prime $p$ between $2n$ and $3n$. I'd like to know whether there is an upper bound on $p$ or whether there is an upper bound on a prime between $2n$ and $kn$, where $k$ is an odd integer.

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closed as not a real question by Andrés E. Caicedo, Pete L. Clark, Qiaochu Yuan Feb 23 '12 at 4:41

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

$3n-1{}{}{}{}$? – anon Feb 23 '12 at 3:10

Your question makes no sense. However...the next prime after some prime $p$ is, on average, approximately $p + \log p.$ Some well known conjectures on large prime gaps, sort of mushed together, suggest that the next prime after $p$ is no larger than $$ p + 3 \; \log^2 p.$$ There is not the slightest hope of proving this, but it holds as far as has been computed, plus it is also true for small primes such as $2.$ See GAPS

Here are some TABLES about large prime gaps. Given that they do not necessarily print out the prime $p$ just before the gap, we can infer, with gap $g,$ that $$ \frac{g}{\log^2 p} = \frac{\mbox{merit}^2}{g}, $$ because $$ \mbox{merit} = \frac{g}{\log p} $$

See, especially, CRAMER_GRANVILLE

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There are a few things that have been proved. E.g., it's known there's a prime between $n$ and $(1.2)n$ for $n$ exceeding some known $n_0$; also, there's a known $c$, $.5\lt c\lt1$, such that there's a prime between $n$ and $n+n^c$ for $n$ sufficiently large. – Gerry Myerson Feb 23 '12 at 6:31

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