Show that if $x,y,z$ are positive integers, then $(xy + 1)(yz + 1)(zx + 1)$ is a perfect square iff $xy +1, yz +1, zx+1$ are all perfect squares.
$(xy + 1)(yz + 1)(zx + 1) $ is a perfect square if and only if each factor is
Disable password request from suspend - 18.04 Steps to solve this system of equations: $\sqrt{x}+y=7$, $\sqrt{y}+x=11$ Imagine I have tons of RAM. Let's say 64GB. That's a lot for even gaming PCs. Now the default location of a pagefile in Windows is on the main OS drive, be it HDD or SSD, which are faster in general, but still not as fast as RAM.  Something tells me that disabling the pagefile on the hard drive or creating a virtual RAM drive and letting the pagefile be there could make Windows move all its virtual memory to RAM, and so increase the system's performance, but I'm not very knowledgeable in that area, so that might not be true at all.  I tried both, but I couldn't analyze the results to reach a definite conclusion with my knowledge level in memory things.  Would this work? If not, why? What I am looking for is the equivalent of System.Windows.SystemParameters.WorkArea for the monitor that the window is currently on.  Clarification:  The window in question is WPF, not WinForm. Should I use UUID as well as ID I'd like to highlight paragraphs such that the highlight color appears as one whole rectangle underlying the entire paragraph. I have looked around but nothing gives me the desired effect so far.  For example \fcolorbox{} from xcolor cannot highlight an entire paragraph; \hl{} from soul leaves out spacings between lines uncolored.  I am wondering how I can do this? At what point is the ~/.bashrc file created? If $F:M\to N$ is a smooth embedding, then so is $dF:TM\to TN$. How to know if a time series is stationary or non-stationary? How to parse a string to an int in C++? What's the C++ way of parsing a string (given as char *) into an int? Robust and clear error handling is a plus (instead of ). Avoiding if statement inside a for loop? Suppose you want to ask someone to do something, say to tell you the way to a station. Which is more polite, "Would you tell me the way to the station?" or "Could you tell me the way to the station?"? On Linux can I keep a program running in background even after I disconnect? I'm trying to install a Matlab R2011a launcher for Unity in Ubuntu 12.04. I've tried  (although I know it's for 11.10 and mentions that even 11.10 is an unsupported OS for Matlab R2011a) but without any satisfactory solution.  This is my launcher file, /usr/share/applications/matlab.desktop:  #!/usr/bin/env xdg-open [Desktop Entry] Type=Application Icon=/usr/share/icons/matlab.png Name=MATLAB R2011a Comment=Start MATLAB - The Language of Technical Computing Exec=matlab -desktop Categories=Development;   I open the dash panel and search for "matlab". This launcher is found among applications. I click it, and Matlab's splash screen shows up, but when it disappears the program doesn't start. (I've verified with htop that no matlab-processes are running in the background either.)  If I add  Terminal=true   to the launcher file, the program starts OK, and opens a terminal as well as Matlab. However, both the terminal and Matlab itself show up in the Launcher area, with the Matlab icon, so it looks like I have two Matlab instances running when really it's only one. (Actually, they show up as two different programs, and not just two instances of the same - the icons are independent, not grouped together.)  This is definitely not optimal. I had hoped to create a launcher I can lock to the launcher area, and then that same icon will be the icon for the active Matlab instance when the program is running.  How do I create a launcher for Matlab that works as expected?  Update: I was apparently a bit unclear on my symptoms, I'll try to clarify a little. I've also tried some suggestions from the answers, and further investigated what happens. My current setup (a launcher file with Terminal=true and Exec=matlab -desktop -nosplash &amp;) renders the following behavior:   I open Dash by pressing the Windows key on my laptop, and search for "matlab". It finds the launcher named "MATLAB R2011a". I click it. A terminal window opens, using the icon I referred to in the launcher file. Almost immediately, MATLAB's splash screen also opens, using the same icon (and thus being grouped with the terminal window in the launcher). The splash screen disappears and, so does one of the icons in the launcher. The MATLAB desktop environment opens, using a different version of the icon which is displayed next to the icon for the terminal window (not grouped with it).   I can lock the terminal window's icon to the launcher and successfully start MATLAB by clicking it, but it doesn't feel optimal that I start the program with one icon, and switch to it with another. I've also tried the following:   Exec without the ampersand &amp; in the launcher command, but it didn't make a difference. Executing matlab -nosplash manually from a terminal still shows the splash screen. (What, then, does the nosplash option really do?) What is the value of the series   $\sum\limits_{n=1}^\infty {\frac{1}{10^n-1}}$   and can one find closed forms for the sums   $\sum\limits_{n=1}^N {\frac{1}{10^n-1}}$?   I found that $\sum_{n=1}^N {\frac{1}{10^n-1}} $  =  $\sum_{n=1}^N (\lim_{m\to\infty  }{\frac{10^{-n}-10^{-nm}}{1-10^{-n}}})$  =  $0.111\ldots+0.0101\ldots+\ldots$ but I can't go farther.     The original question is: "Where does the second '1' appear?" Short connection at IAD (Dulles international airport) I am flying from New Orleans to Dulles (IAD) on United Express, and from IAD to Frankfurt on United. I have a short connection (45 minutes) in IAD.   I'm trying to figure out how can I make it:  Will I need to go throughout security and border check since I'm leaving the US? How big is IAD airport? and are there different terminals for domestic and international flights? If so, how long would it take to walk between them? $\sum_{n=1}^\infty\frac{z^n}{n}$ does not converge uniformly on $\mathbb{D}$. A phonon is a quantized unit of sound; they are encountered when quantizing lattice vibrations in solids. Now, even an ideal gas supports sound waves, but in this case, interactions between atoms are weak. That makes it hard to imagine what a quantized vibration would look like, since at small scales, the particles are free!   Is there a phonon picture for sound in an ideal gas? Is it ever useful? Why does the version of Java verified by Oracle differ from the version shown at the command line? Number of vertices in the Klein bottle triangulation given in "Topology and geometry for physicists" by Nash and Sen I stumbled upon the following triangulation of the Klein Bottle in page 76 of the book Topology and geometry for physicists by Charles Nash and Siddhartha Sen.    Below the figure, it says "the triangulation $K$ consists of eight $0$-simplexes[...]".  I think this is not an eight-vertex triangulation, but rather a nine-vertex triangulation of the Klein bottle. I also know, thanks to the great comments under  question, that an $8$-vertex triangulation of the Klein bottle is a minimal one, hence the one portrayed in the book isn't minimal.  Am I correct?