Let $p_n$ be the $n$-th prime. Is there an elementary way to prove the statement that there exists a polynomial function $P$ such that $$p_n \leq P(n),$$ for all $n \in \mathbb{N}$? (Or, alternatively, for all $n$ from some $N \in \mathbb{N}$ onward?) I know the statement is true, and that $p_n$ &quot;behaves asymptotically like&quot; $n \log n,$ but I'm specifically looking for an elementary proof. (I'd like to show $\sqrt[n]{p_n} \to 1,$ and since trivially $n \leq p_n,$ the result would follow from the squeeze theorem.) Thanks in advance!!!!!!
Is there a way to show that $\sqrt{p_{n}} &lt; n$?  In , I show that $f_{2}(x)=\frac{x}{ln(x)} - \sqrt{x}$ is ascending, for $\forall x\geq e^{2}$. As a result, $\forall n \geq 3$ $$\frac{p_{n}}{ln(p_{n})} - \sqrt{p_{n}}\leq \frac{p_{n+1}}{ln(p_{n+1})} - \sqrt{p_{n+1}}$$ Also (and as a result), $\forall n \geq 3$ $$ \frac{p_{n}}{ln(p_{n})} - \sqrt{p_{n}} &gt; 0$$ Or $$ \frac{\pi (p_{n})}{p_{n}/ln(p_{n})} &lt; \frac{\pi (p_{n})}{\sqrt{p_{n}}}$$  According to  $$\displaystyle\smash{\lim_{n \to \infty }}\frac{\pi (p_{n})}{p_{n}/ln(p_{n})}=1$$ Or, $\forall \varepsilon &gt;0$, $\exists N(\varepsilon )$: $\forall n&gt;N(\varepsilon )$ $$1- \varepsilon &lt; \frac{\pi (p_{n})}{p_{n}/ln(p_{n})} &lt; 1+ \varepsilon$$ Or $$1- \varepsilon &lt; \frac{\pi (p_{n})}{p_{n}/ln(p_{n})} &lt; \frac{\pi (p_{n})}{\sqrt{p_{n}}}$$ As a result $\forall \varepsilon &gt;0$, $\exists N(\varepsilon )$: $\forall n&gt;N(\varepsilon )$ $$(1 - \varepsilon ) \cdot  \sqrt{p_{n}} &lt; \pi (p_{n}) = n$$  But this is not enough.  Interestingly,  is true iff function $f_{4}(x)=\pi (x) - \sqrt{x}$ is strictly ascending ($x &lt; y \Rightarrow f(x) &lt; f(y)$) for prime arguments.  If $f_{4}(p_{n}) &lt; f_{4}(p_{n+1})$ then $$\pi (p_{n}) - \sqrt{p_{n}} &lt; \pi (p_{n+1}) - \sqrt{p_{n+1}}$$ Or $$\sqrt{p_{n+1}} - \sqrt{p_{n}} &lt; \pi (p_{n+1}) - \pi (p_{n}) =1$$  And vice-versa, if $$\sqrt{p_{n+1}} - \sqrt{p_{n}} &lt; 1$$ Then $$-\sqrt{p_{n}} &lt; -\sqrt{p_{n+1}} + 1$$ Or $$\pi (p_{n})-\sqrt{p_{n}} &lt; \pi (p_{n}) + 1 -\sqrt{p_{n+1}} = \pi (p_{n+1}) -\sqrt{p_{n+1}}$$  So, if Andrica's conjecture is true then $\forall n \geq 3$ $$\pi (p_{n})-\sqrt{p_{n}} &gt; 0$$ Or $$\sqrt{p_{n}} &lt; \pi (p_{n})= n$$
garbled xml output How do I make protocol foo hostname-aware? This is a  about hostname-awareness and proxying.   I know that some protocols are hostname-aware; that is, when I connect to the HTTP server at www.example.com it knows I want www.example.com's HTTP service, not www.example.net's, even though they are on the same IP address.  How can I do this for protocol foo?  (Temporary note: this question appears pursuant to .) How to filter a list by hour Let $a,b,c $ are non-negative real numbers, and $a+b+c=3$.  How to prove inequality $$ ab^2+bc^2+ca^2\le 4.\tag{*} $$    In other words, if $a,b,c$ are non-negative real numbers, then how to prove inequality $$ 27(ab^2+bc^2+ca^2)\le 4(a+b+c)^3.\tag {**} $$ $\color{gray}{\mbox{(Without using "universal" Lagrange multipliers method).}}$  Thanks! How do they express the time, in American and British English? Integer partitions in all orderings On which Stack Exchange site can I ask questions like        When will [some new feature] for [some product] be released?   Where can I download an update for [some product]?   I'm getting errors when accessing the customer portal for [some company] – how can I fix this?   Why was I banned for doing [some action] on [some platform]?   Why is [some company] discontinuing [some product]?   Why is [some company] giving away [some product] for free?   What's the price for [some product]?      or any other product support/customer service questions?  It's not a product related to programming (or I've read  on Meta Stack Overflow) so I know Stack Overflow isn't the right place. Which other site in the network will take these kind of questions? Integrate $\int \frac{25 - x^2}{x^4} \operatorname d\!x$ I think we use trig substitution for this. But I'm not sure where to even begin with this   $$\int \frac{25 - x^2}{x^4} \operatorname d\!x$$ Desktop icons rearrange after using remote desktop connection I sometimes remote into my desktop PC (1900x1280) from my laptop (1280x900).  When I do, Windows (Vista) on the desktop PC rearranges all my desktop icons.  I could understand this if the RDP session's desktop size was too small to fit them in, but it's not.  I have about 12 icons, which do fit on the laptop's resolution (I've tried).  Why would Windows rearrange them all like this each time? In this case I'm filtering by the tag [tag:css*], but it's showing this as [css*] and showing the  tag's exerpt. Strange GET requests in logs I'm getting the following GET requests in my apache logs:  109.230.251.14 - - [29/Mar/2011:16:28:18 +0100] "GET http://209.191.92.114/config/pwtoken_get?login=jackmcphee232&amp;src=ygodgw&amp;passwd=e59e2240415e6f1aba3da72b8f189f4e&amp;challenge=9TbU_9yfZhKmzlHtK9X4OkQlesTH&amp;md5=1 HTTP/1.0" 404 1226 "-" "-"   Any idea what it could be and how is that possibly a get request? That IP address seems to point to Yahoo! I'm very confused. References in externalized pgfplots I was writing on an document with a lot of pgfplots in it. Because of the memory limitation in LaTeX and in order to save some compiling time I tried out the externalize command. With most of the plots it perfectly worked out. But when I use the \ref command, namely \ref{&lt;legendname&gt;} the legend gets lost and instead there is a ??.  I think this question in a way is similar to  and . But no answer to these questions could solve my problem. Even the use of xr and zref packages showed no effect.  Am I just missing something?    MWE of the maindocument:  \documentclass{article} \usepackage{zref-user} \usepackage{xr-hyper} \usepackage{pgfplots} \usetikzlibrary{pgfplots.groupplots} \usepgfplotslibrary{external}  \tikzexternalize \externaldocument{part1}  \begin{document} \input{part1} \end{document}     With part1.tex looking like this:  \begin{tikzpicture}     \begin{groupplot}[       group style= {  columns=3,xlabels at=edge bottom,                 y descriptions at=edge left,                 horizontal sep=2cm,group name=plots},     width=0.4\textwidth,     xtick=data,     legend columns=3]     \nextgroupplot[xlabel=site 1,legend to name=grouplegend]     \addplot coordinates {(1,20)(2,23)(3,24.5)};     \addlegendentry {AaAaA,BbBbB,CcCcC}%     \nextgroupplot[xlabel=site 2]     \addplot coordinates {(1,14)(2,16)(3,17)};     \nextgroupplot[xlabel=site 3]     \addplot coordinates {(1,10)(2,11.5)(3,12.25)};     \end{groupplot}   % Legend     \node at (plots c2r1.north) [anchor=south] {\ref{grouplegend}}; \end{tikzpicture}     I'm writing with MikTeX 2.9 and the latest updates of the packages used. For compiling I used the command pdflatex -enable-write18 &lt;file name&gt;.  I hope the MWE is short enough ;) and some knows how to solve this problem. Why does inertia happen? In  R P Feynman relates a story where his father told him that, even though we know the word "inertia" and what it means, nobody knows why inertia happens. Is that still true? How can I write the name show in axis of 3D array using TiKz I am drawing a 3D array like the attached picture, which actually describes a multidimensional array. However, I can not put the name of the axis and take the value from the matrix like the attached picture.  This is the code that I have:  \documentclass[margin=5mm]{standalone} \usepackage{tikz} \usetikzlibrary{matrix,calc}  \begin{document} \begin{tikzpicture}[every node/.style={anchor=north east,fill=white,minimum width=1.4cm,minimum height=7mm}] \matrix (mA) [draw,matrix of math nodes] { (1,1,3) &amp; (1,1,3) &amp; (1,1,3) &amp; (1,1,3) \\ (1,1,3) &amp; (1,1,3) &amp; (1,1,3) &amp; (1,1,3) \\ (1,1,3) &amp; (1,1,3) &amp; (1,1,3) &amp; (1,1,3) \\ (1,1,3) &amp; (1,1,3) &amp; (1,1,3) &amp; (1,1,3) \\ };  \matrix (mB) [draw,matrix of math nodes] at ($(mA.south west)+(1.5,0.7)$) { (1,1,3) &amp; (1,1,3) &amp; (1,1,3) &amp; (1,1,3) \\ (1,1,3) &amp; (1,1,3) &amp; (1,1,3) &amp; (1,1,3) \\ (1,1,3) &amp; (1,1,3) &amp; (1,1,3) &amp; (1,1,3) \\ (1,1,3) &amp; (1,1,3) &amp; (1,1,3) &amp; (1,1,3) \\ };  \matrix (mC) [draw,matrix of math nodes] at ($(mB.south west)+(1.5,0.7)$) { (1,1,3) &amp; (1,1,3) &amp; (1,1,3) &amp; (1,1,3) \\ (1,1,3) &amp; (1,1,3) &amp; (1,1,3) &amp; (1,1,3) \\ (1,1,3) &amp; (1,1,3) &amp; (1,1,3) &amp; (1,1,3) \\ (1,1,3) &amp; (1,1,3) &amp; (1,1,3) &amp; (1,1,3) \\ };  \draw[dashed](mA.north east)--(mC.north east); \draw[dashed](mA.north west)--(mC.north west); \draw[dashed](mA.south east)--(mC.south east); \end{tikzpicture}   \end{document} Non-Exchange address in Exchange GAL We just made the switch to Google Apps for part of our domain. We disabled the accounts that are now on Google Apps and mail routes correctly. Our problem is that now we have no way of seeing those users in the Global Address List.  So we have these users in our Active Directory, they have no mailbox on Exchange, and we want to be able to see/search for their addresses in the GAL. Is there any way to do this?  This is Exchange 2007, if that matters. Opposite of Alternatives Are bridges like hubs or switches? It is my understanding that the main difference between a network hub and a network switch is that the hub always sends out all packets to all (other?) hosts while the switch is smart about it and keeps track of which port belongs to which MAC address via MAC address tables and whatnot. If a packet is addressed to a host which is connected to the switch but have never sent or received anything, the switch acts like a hub would. But every subsequent packet addressed to that host is not broadcast to the other hosts but forwarded only to that specific host. Please correct me if I got something wrong. Does a network bridge in Linux act like a hub would, or like a switch? Join CSV file to shapefile I wanted to join a CSV file to a shapefile.  When I identified a field to facilitate the join, e.g.,   map &lt;- spChFIDs(map, as.character(map$ID))   it returned     Error in spChFIDs(SP, x) : lengths differ   Can anyone advise? It seems that a common operation needed when defining new commands is to scan over a list of arguments (separated e.g. by a comma, but maybe by something else) and do something with each argument in the list, find the last one, or whatever.  For example to write commands that look like \Command{a,b,c} or \Command{c.d.e} and do something with each a, b, and c. See e.g. questions on  and .  Some solutions have been posted there, but what would be the best idiom to perform such kind of operations? LaTeX's \@for looks very nice, but is it possible to use it to scan anything other than comma-separated lists? Is time continuous or discrete?