Prove that if m is an integer and there is a rational number r/s so that (r/s)^2 = m, then there is an integer n so that n^2 = m.
How to prove: if $a,b \in \mathbb N$, then $a^{1/b}$ is an integer or an irrational number?
$\gcd(a,b)=1, x^a = y^b\Rightarrow x = n^b$, $ y = n^a$ for an integer $n$. If $ a$, $ b$, $ x$, $ y$ are integers greater than $1$ such that $ a$ and $ b$ have no common factor except $1$ and $ x^a = y^b$ show that $ x = n^b$, $ y = n^a$ for some integer $ n$ greater than $1$.  I started this way: $x^a=y^b$ $=&gt;$ $x=y^{b/a}=(y^{1/a})^b$. Thus $x$ is an integer, and so is $y^{1/a}$, so $y=n^a$ for some integer $n$ greater than $1$. Then from the given relation, we get $x=n^b$.   I am not very convinced with my solution, as I feel I did some mistake. Can anyone point out where the mistake is? Any other solution is also welcome :) How can I get free or trial ExactTarget instance I recently made changes to my fstab file and, upon reboot, noticed I had an error in it. I get the typical mount error on boot. I went to try and modify it by entering the read-only mode but then, since it's read only, I can't access it. Is there a way around this or some way to delete the file so I can use the backup (which I'm pretty sure works)? There is a nice property of normal spaces, namely, closed disjoint subsets can be separated by continuous functions into $\mathbb R$.   Then you ask yourself, what about Hausdorff spaces?, are all Hausdorff spaces completely Hausdorff?   Well, if we choose our separating space $X$, that is, the space such that for any Hausdorff $Y$ and any distinct $a,b\in Y$ there is some continuous $f:Y\rightarrow X$ with $f(a)\neq f(b)$, to be completely regular and Hausdorff, this cannot happen by an example of E. Hewitt, Hewitt’s Condensed Corkscrew, call it $\mathbb H$; as any continuous function from $\mathbb H$ into $\mathbb R$ is constant, and any Hausdorff completely regular space $X$ can be embedded into a product of closed intervals $\prod_{i\in I}[0,1]$, and thus any continuous $f:\mathbb H\rightarrow X\subseteq \prod_{i\in I}[0,1]$ is constant, as each $\pi_i\circ f$ is constant, so our candidate to be the separating space $X$ cannot even be $[0,1]$.  My question is, does there exist a Hausdorff space $X$ such that for all Hausdorff spaces $Y$, and any distinct $a,b\in Y$ there exists a continuous $f:Y\to X$ such that $f(a)\neq f(b)$?  Thanks. Does anybody know how to easily undo a git rebase?  The only way that comes to mind is to go at it manually:   git checkout the commit parent to both of the branches  then create a temp branch from there cherry-pick all commits by hand replace the branch in which I rebased by the manually-created branch   In my current situation this is gonna work because I can easily spot commits from both branches (one was my stuff, the other was my colleague's stuff).  However my approach strikes me as suboptimal and error-prone (let's say I had just rebased with 2 of my own branches).  Any ideas?  Clarification: I'm talking about a rebase during which a bunch of commits were replayed. Not only one. Trying to identify pre-1970s short story involving multiple humans absorbed into an alien blob I'm playing around with a native (non-web) single-player game I'm writing, and it occured to me that having a daily/weekly/all-time online highscore list (think Xbox Live Leaderboard) would make the game much more interesting, adding some (small) amount of community and competition. However, I'm afraid people would see such a feature as an invitation to hacking, which would discourage regular players due to impossibly high scores.  I thought about the obvious ways of preventing such attempts (public/private key encryption, for example), but I've figured out reasonably simple ways hackers could circumvent all of my ideas (extracting the public key from the binary and thus sending fake encrypted scores, for example).  Have you ever implemented an online highscore list or leaderboard? Did you find a reasonably hacker-proof way of implementing this? If so, how did you do it? What are your experiences with hacking attempts? Can an irrational number raised to an irrational power be rational? From , I have tried the using \hspace{0pt} and   \makeatletter \renewcommand{\dddot}[1]{%   {\mathop{\kern\z@#1}\limits^{\vbox to-1.4\ex@{\kern-\tw@\ex@    \hbox{\normalfont ...}\vss}}}} \renewcommand{\ddddot}[1]{%   {\mathop{\kern\z@#1}\limits^{\vbox to-1.4\ex@{\kern-\tw@\ex@    \hbox{\normalfont....}\vss}}}} \makeatother   after loading mathtools but the problem still persists.    \documentclass{article} \usepackage{mathtools} \begin{document} \[ \dot{\mathbf{r}}\times\ddot{\mathbf{r}}\times\dddot{\mathbf{r}} \] \end{document}   In the case the font scheme I am using has any affect as well, the fonts are:  \usepackage[utf8]{inputenx} \renewcommand{\rmdefault}{ppl}                  \linespread{1.05}                   \usepackage[scaled]{helvet} \usepackage{courier}               \usepackage{eulervm} \normalfont       \usepackage[T1]{fontenc}            \usepackage{textcomp} Unable to cover get set in apex test class How do you change your hostname in Windows 7? Not the computer name, but the hostname. Looking at moving my laundry upstairs, and the right answer is to combine two adjacent closets to get the width and move one of their walls a foot or two into the kitchen to get the depth.  The kitchen floor is 10-inch-square tiles. It would be sort of nice if that continued on into the laundry closet. I do have a full bundle of spare tiles in the basement.   Question: Could/should I plan on the new wall coming in on top of the existing tiles (presumably having to drill to secure the footer without shattering them), or should I plan on removing the ones under the footer and fitting cut pieces back in afterward? Is there a rule of thumb for objective questions asked out of curiosity? Isaac Moses, the patriarch of Mi Yodeya, :  [...] the general Stack Exchange directive, expressed in the uneditable part of that section of the About page, to &quot;Focus on questions about an actual problem you have faced. Include details about what you have tried and exactly what you are trying to do.&quot; In fact, while this directive is perfect for the behemoth programming site that Stack Exchange grew out of, it doesn't fit very well for any of the more intellectually-oriented, rather than practically-oriented sites. [...]  This is something that's bugged me for a while, and ended up prompting a discussion between Anna, Aarthi and me. The gist of it: is there any simple test that can be used to identify answerable questions asked in good faith when the &quot;problem to be solved&quot; is... simple curiosity? Obviously, this is something that sites like  and  have had to contend with for some time, but now and then such questions pop up even on sites where folks are generally trying to solve real problems. While a potentially interesting thought experiment, . Strictly-speaking, problems like that fail the  test. And yet, I would keep that question simply because I learned something useful from the answers − not necessarily something I would ever apply to socks mind you, but certainly something I might apply to similar problems. I don't believe the question was asked in bad faith or for mindless social fun; yet, I have a bit of trouble stretching the definition of &quot;based on actual problems&quot; to contain it. Obviously, the fact that we're hosting a growing number of sites dedicated mostly or entirely to intellectual curiosity makes this something of a rhetorical question. The &quot;actual problems&quot; part of the FAQ was added specifically to draw a line between good and bad  − heading off  and their ilk. But it's a pretty damn good litmus test, and if it's ok to ignore it for some questions − or particularly for  − then what should stand in its place? Python analog of PHP's natsort function (sort a list using a "natural order" algorithm) Where can I see my flag count / history? Is there a section or page that lists my flag count and / or history? How do I take a screenshot on my Mac? What's the easiest way to take a screenshot of my Mac?  I've used Grab.app, but it's a bit annoying just to get a quick screenshot. Is there a faster way? Is it possible to adjust desktop monitor brightness like on laptops? Yes all desktop monitors have separate menu for it. But is it possible to change this to something like Winkey+(F1..F12)?    Monitor is connected via VGA or DVI cable.   OS: Ubuntu 14.04 Desktop monitors Bonferroni adjustment in SPSS - what does it do? Splitting all lines in layer at vertices using QGIS I'm trying to find a discontinuous linear functional into $\mathbb{R}$ as a prep question for a test. I know that I need an infinite-dimensional Vector Space. Since $\ell_2$ is infinite-dimensional, there must exist a linear functional from $\ell_2$ into $\mathbb{R}$. However, I'm having trouble actually coming up with it.   I believe I'm supposed to find an unbounded function (although I'm not sure why an unbounded function is necessarily not continuous; some light in that regard would be appreciated too), so I thought of using the vectors $e^i$, which have all entries equal to zero, except for the $i$-th one. Then, you can define $f(e^i)=i$.  That'd be unbounded, but I'm not sure if it'd be linear, and even if it is, I'm not sure how to define it for all the other vectors in $\ell_2$.   A friend mentioned that at some point the question of whether the set $E=\{e^i:i\in\mathbb{Z}^+\}$ is a basis would come up, but I'm not sure what a basis has to do with continuity of $f$.   I'm just learning this topic for the first time, so bear with me please.   The space of sequences that are eventually zero (suggested by a few people) turned out to be exactly what I needed.  It also helped to cement the notions of Hamel basis, not continuous, etc.