# Is there an upper bound on a prime between $2n$ and $3n$? [closed]

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

## closed as not a real question by Andres Caicedo, Pete L. Clark, Qiaochu YuanFeb 23 '12 at 4:41

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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 $$\mbox{merit} = \frac{g}{\log p}$$
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