I found a way to prove this using Chebychev's theorem, are there ways to solve it without relying on this?
If $n&gt;1$ is an integer, then $\sum \limits_{k=1}^n \frac1k$ is not an integer.  If you know , then you know there must be a prime $p$ between $n/2$ and $n$, so $\frac 1p$ appears in the sum, but $\frac{1}{2p}$ does not. Aside from $\frac 1p$, every other term $\frac 1k$ has $k$ divisible only by primes smaller than $p$. We can combine all those terms to get $\sum_{k=1}^n\frac 1k = \frac 1p + \frac ab$, where $b$ is not divisible by $p$. If this were an integer, then (multiplying by $b$) $\frac bp +a$ would also be an integer, which it isn't since $b$ isn't divisible by $p$.  Does anybody know an elementary proof of this which doesn't rely on Bertrand's Postulate? For a while, I was convinced I'd seen one, but now I'm starting to suspect whatever argument I saw was wrong.
There exists $C\neq0$ with $CA=BC$ iff $A$ and $B$ have a common eigenvalue Question:  Suppose $V$ and $W$ are finite dimensional vector spaces over $\mathbb{C}$. $A$ is a linear transformation on $V$,  $B$ is a linear transformation on $W$. Then there exists a non-zero linear map $C:V\to W$ s.t. $CA=BC$ iff $A$ and $B$ have a same eigenvalue.  ===========This is incorrect==========  Clearly, if $CA=BC$, suppose $Ax=\lambda x$, then $B(Cx)=CAx=C(\lambda x)=\lambda (Cx)$, so $A$ and $B$ have same eigenvalue $\lambda$. On the other hand, if $A$ and $B$ have a same eigenvalue $\lambda$, suppose $Ax=\lambda x, By=\lambda y$. Define $C:V\to W$ s.t. $Cx=y$, then $BCx=By=\lambda y=C\lambda x=CAx$. But I don't kow how to make $BC=CA$ for all $x\in V$.   ====================================== I want to write several classes, most of them will use KOMA classes as base class.  However I stumbled upon the problem that some options (mainly concerned with layout) are not processed if given in the wrapper class:    \begin{filecontents}{test.cls}     \ProvidesClass{test}[2015/01/01]     \DeclareOption*{\PassOptionsToClass{\CurrentOption}{scrartcl}}     \ProcessOptions\relax     \LoadClass[paper=a5]{scrartcl}   \end{filecontents}  \documentclass{test}  \usepackage{blindtext}  \begin{document}   \section{Test}   \blindtext \end{document}   This results in   LaTeX Warning: Unused global option(s):     [paper=a5]  If i give the option as \documentclass[paper=a5]{test} it is used correctly, because I pass the options to scrartcl. However I want to change the defaults for the layout.   How can I achieve this? Is short-circuiting logical operators mandated? And evaluation order? Does the ANSI standard mandate the logical operators to be short-circuited, in either C or C++?  I'm confused for I recall the K&amp;R book saying your code shouldn't depend on these operations being short circuited, for they may not. Could someone please point out where in the standard it's said logic ops are always short-circuited? I'm mostly interested on C++, an answer also for C would be great.  I also remember reading (can't remember where) that evaluation order isn't strictly defined, so your code shouldn't depend or assume functions within an expression would be executed in a specific order: by the end of a statement all referenced functions will have been called, but the compiler has freedom in selecting the most efficient order.  Does the standard indicate the evaluation order of this expression?  if( functionA() &amp;&amp; functionB() &amp;&amp; functionC() ) cout&lt;&lt;"Hello world"; LLC-SCorp to self - actually, does the corp pay payroll taxes? In the USA. Here are the two common scenarios for the self-employed:  A. Joe makes 100,000 over the year. It's paid to him as 1099 payments from his various clients.  Joe pays the many taxes on that.  B. Joe makes 100,000 over the year.  He has an LLC as SCorp (or maybe an SCorp per se).  The clients pay $100,000 to the entity JoeCorp.  JoeCorp pays Joe a certain salary (it might be the whole 100,000, or, it might be (let's say) half of that and the rest is taken as profit distribution, not relevant here).  So, JoeCorp (his LLC) pays Joe say $50,000.  In fact: does JoeCorp have to pay "payroll taxes" on that??  Does it perhaps make a difference if it is an LLC with S-Corp election v. S-Corp?  (I realize there are many questions on here about "1099 v. corp" .... but I've never really understood this and can't seem to google it up.) I know that a stationary metric is one which possesses a timelike Killing Vector (so I have time invariance).  I read that a static metric is one which has a timelike Killing vector (so it is stationary) AND this Killing vector is orthogonal to a family of hypersurface (of dimension n-1 is n is the dimension of my manifold). I don't understand the "physical" meaning behind this definition ? Is there a way to explain this in some more intuitive way ? Thanks Microphone Not Working With FaceTime Problem on Sum of Digits of a Number I was given this problem to Prove if $Q(n)=Q(2n)$ implies $9∣n$. Where $Q(n)$ be the sum of digits of $n$. I tried to use the fact that sum of digits is congruent to the number modulo $9$ but I had no luck. Any help is appreciated. Book recommendations lists Ubuntu 4.10 (Warty Warthog) as the first Ubuntu released by Canonical Ltd.  But I wanted to know whether there ever was an Ubuntu 1.x, as in Windows or Mac OS. If yes, is it available somewhere? Remove "Show Desktop" button next to the clock in Windows 7 Which other nations are there in Attack on Titan world aside Eldia and Marley? In the Attack on Titan wikia they often mention other nations aside Eldia and Marley which supposedly have more advanced technology than this 2 nations. Which other known nations are there in Attack on Titan world aside Eldia and Marley and why are they more advanced than Marley when only Eldia is supposed to be isolated from the world? Let $n \geq 1$ be an odd integer. Show that $D_{2n}\cong \mathbb{Z}_2 \times D_n$. Let $n \geq 1$ be an odd integer. Show that $D_{2n}\cong \mathbb{Z}_2 \times D_n$. I define a map $$\phi:D_{2n} \rightarrow \mathbb{Z}_2 \times D_n$$ by $R \mapsto (0,r^{\frac{n+1}{2}})$ and $M \mapsto(1,m)$. Then I am stuck at showing the map is bijective. By the way, do we need to show that $(0,r^{\frac{n+1}{2}})(1,m)(0,r^{\frac{n+1}{2}})^{-1}=(1,m)$ ? If we can prove this, then what can we conclude? Accidently deleted boot option on Dell laptop "See somebody do" and "see somebody doing" Should I use "who" or "that" when referring to an organisation? I generally use the rule of thumb of using who when referring to a person and that when referring to an object. Example:  Jim is someone who makes me smile. I was nearly knocked out by the ball that hit me.  But what about when we're talking about an organisation?  I am forever indebted to the charities who helped me. I am forever indebted to the charities that helped me.  The former seems somewhat warmer, if that makes sense, although the latter feels more correct. What should I use? And how about this:  I'm really happy with the insurance company who helped me with my claim. I can't stand them, they're the insurance company that ripped me off.  To me the former evokes a team of helpful people while the latter evokes a cold heartless corporate entity. Is this just a matter of style and so should go with my instincts, or should I aim for what is 'correct?' Frequent Write to Non-volatile Memory I can't run this latex script How is the authenticity of Debian packages guaranteed? Is it possible to flash a smartphone like the Samsung Galaxy SIII or an LG Optimus 4X with a vanilla Android ROM like the one that has been flashed on Galaxy Nexus or Motorola phones (I think)?  I am asking because manufacturers like Samsung, LG, HTC tend to make ROMs for their smartphones and tablets with preinstalled launchers like Touchwiz or HTC Sense. They explain their decision by saying that they want to make their devices to look unique against their rival devices. The problem is that Android updates for their phones and tablets get delayed and there are incidents in which some devices - which are more than capable of running the upcoming version of Android - fall way behind or never get updated at all.   So, what I am basically asking here is if there is a way to install vanilla Android, excluding CyanogenMod or any other modified vanilla ROMs, on an Android based smartphone? In a project the customer wants the images of the WMS to be secured against unauthorized access.   What are the common solutions to realize that?   Are they compatible with the OGC-standards?