How would I prove the following:      Show that an odd prime $p$ can be written on the form $p=x^2+2y^2$ for some $x,y\in\mathbb Z$ iff $p\equiv_8 1, 3$. Hint: use the quadratic reciprocity and the fact that $\mathbb Z[\sqrt{-2}]$ is a Euclidean domain.
For a prime $p \equiv 1$ or $3$  (mod $8$), show that the equation $x^{2} + 2y^{2} = p$ has a solution.   I'd like to see an elementary proof.
How do I use 'chmod' on an NTFS (or FAT32) partition? Python Referenced For Loop $TT^*=T^2$, show that $T$ is self-adjoint Let $V$ be an inner product space, finitely generated over $\mathbb{C}$, $T\in \operatorname{End}(V)$ that satisfies $TT^*=T^2$, show that $T$ is self-adjoint.    I know that $TT^*$ is positive so has positive square-root, thus the square root is positive definite. But I don't feel like this is a proof. Can anyone prove it without considering square root? How does rooting work? Or, how do I root my device without a rooter?  Rooting instructions for Android devices that I have seen are typically of the following form:   Download Gingerbreak.apk Run Gingerbreak, click “root device” Observe that Superuser.apk has been magically installed, possibly along with busybox and other things, and enjoy your rooty goodness.   What I am wanting to know is this: how does the rooting actually happen? What does Gingerbreak do?  I understand the end result of rooting:   su is present and working Superuser.apk is installed and controls access to su Various utilities are present, provided by BusyBox (optional?)   I also think I understand the basic idea of how Gingerbreak gets started — it exploits some local privilege escalation vulnerability and uses its new-found powers to deploy Superuser and whatever else is necessary.  So, unless I am wrong in some of the understanding I laid out above, the core of thing I want to know is what the root installer does to deploy and activate Superuser. It seems that just putting Superuser.apk in the right place wouldn't be enough — how does it get allowed root access? How does it get hooked up to moderate that access for other applications?  What does Gingerbreak do post-exploit to enable and secure root access? How to burn a Windows .iso to a USB device? When you have a link from lets say twitter that is t.co/foo that links back to your website how does that effect SEO and how do the major search engines handle Shortened URLs? What is the name for these type of images? I love these kind of graphics and it's so flat which totally matches my current product development.      Do we have a name for it? A name will make it easier to search for examples and how to's. What is the first movie to show this hard work message at the very end? What is the first movie to show this hard work message at the very end? Why is 'ls' suddenly wrapping items with spaces in single quotes? I just noticed that on one of my machines (running Debian Sid) whenever I type ls any file name with spaces has single quotes surrounding it. I immediately checked my aliases, only to find them intact. wyatt@debian630:~/testdir$ ls 'test 1.txt'  test1.txt wyatt@debian630:~/testdir$ alias alias ls='ls --color=auto' alias wget='wget --content-disposition' wyatt@debian630:~/testdir$   Another test, with files containing single quotes in their names (also answering a request by jimmij): wyatt@debian630:~/testdir$ ls 'test 1.txt'  test1.txt  'thishasasinglequotehere'\''.txt' wyatt@debian630:~/testdir$ touch &quot;'test 1.txt'&quot; wyatt@debian630:~/testdir$ ls ''\''test 1.txt'\'''  test1.txt 'test 1.txt'          'thishasasinglequotehere'\''.txt'   update with new coreutils-8.26 output (which is admittedly much less confusing, but still irritating to have by default). Thanks to Pádraig Brady for this printout: $ ls &quot;'test 1.txt'&quot;   test1.txt 'test 1.txt'    &quot;thishasasinglequotehere'.txt&quot;  $ ls -N 'test 1.txt'  test1.txt test 1.txt    thishasasinglequotehere'.txt  Why is this happening? How do I stop it properly? To be clear, I myself set ls to automatically color output. It just never put quotes around things before. I'm running bash and coreutils 8.25. Any way to fix this without a recompile? EDIT: Appears the coreutils developers  to break with the convention and make this the global default.  UPDATE - October 2017 - Debian Sid has re-enabled the shell escape quoting by default.  And at the bottom of the reply chain to the previous bug report, &quot;the change was intentional and will remain.&quot;  I thought this had already been settled, but apparently it was just reverted so that the &quot;stable&quot; Debian branch could keep its &quot;feature freeze&quot; while getting the other fixes, etc. from the newer version. So that's a shame (in my opinion). UPDATE: April 2019: Just found a  that was caused by this change to ls. When you're confusing developers and generating false bug reports, I think it might be time to re-evaluate your changes. Update: Android toybox ls is now doing something similar to this but with backslashes instead of quotes. Using the -q option makes spaces render as 'question mark characters' (I have not checked what they are, since they're obviously not spaces), so the only fix I have found so far without rooting the device in question is to add this to a script and source it when launching a shell. This function makes ls use columns if in a terminal and otherwise print one-per-line, while tricking ls into printing spaces verbatim because it's running through a pipe. ls() {     # only way I can stop ls from escaping with backslashes     if [ -t 1 ]; then         /system/bin/ls -C $@ |cat     else         /system/bin/ls $@ |cat     fi } Run a command on all subfolders If you have a series of subfolders (like from a to z) and want to run a command on each one of them (like rm *.pdf or ls *.pdf), how do you do that? The "manual" approach would be cd a, rm *.pdf, or ls *.pdf, cd .., cd b, ... That seems too complicated, so I believe there must be an easier approach. No chatoutput in tellraw onClick Event I'm using a /tellraw command to display some kind of menu to the player. How can I react to the clickEvent without posting something in the chat? When I try using the scoreboard, I get an output in my chat like this Set score of &lt;objective&gt; for player &lt;playername&gt; to &lt;value&gt;. The same thing happens, when trying /setblock, /entitydata or /trigger Is there any way, I can execute a command which commandblocks can detect without writing a success/fail message in the chat?  the only commands without a "real" response message I found were /say, /tell and /tellraw. But all of these are impossible to detect(also, they write something else in the chat). Show that an algebraically closed field must be infinite.    Answer  If F is a finite field with elements $a_1, ... , a_n$ the polynomial $f(X)=1 + \prod_{i=1}^n (X - a_i)$ has no root in F, so F cannot be algebraically closed.  My Question  Could we not use the same argument if F was countably infinite? Couldn't we say that if F was a field with elements $a_1, a_2, ... $ then the polynomial $f(X) = 1 + \prod_{i=1}^{\infty} (X - a_i)$ does not split over F?  Thank you in advance Will airlines compensate me if my entertainment system is not working? In long haul flights, if I get a seat and after take off I found out that entertainment system is not working, I asked a flight attendant to change my seat but she apologized because there are no empty seats to move to.   Is it common to ask for a compensation for such an inconvenience? do major airlines compensate you in such cases? I'm a happy Minecraft player. I place my blocks, mine my coal, and generally nobody gets hurt. Lately though, those mean creepers have been sneaking up on me from behind, and killing me. Why? Because my framerate is so darn low.  Adjusting the graphics option (Fancy to Fast) and the render distance help, but they are not enough. I have seen mention of outside optimizations and programs that people have made/used.  What hacks, tricks, and mean mods can I use to improve my Minecraft FPS? I'm looking for an understandable proof of this theorem, and also a complex one involving beautiful math techniques such as analytic number theory, or something else. I hope you can help me on that. Thank you very much How to emulate titletoc with tocloft I was using titletoc until I realized that it has some incompatibilities with hyperref. The thing is, that titletoc can do the same thing in a easier way, less one.  \titlecontents*{subsection}[5.4em] {\small} {\thecontentslabel~ } {} {} [.~\textbullet\ ] [.]   this part does the subsection as a paragraph (with section number and without page number). I have realized that the problem is the pseudo-code used by tocloft for the entries in the TOC  {\cftXfont {\cftXpresnum SNUM\cftXaftersnum\hfil} \cftXaftersnumb TITLE}% {\cftXleader}{\cftXpagefont PAGE}\cftXafterpnum\par   I mean, the last command \par make impossible to leave two subsection entries in the same textline. Any idea? Another interesting solution is with etoolbox, some examples with that package are awesome, but it's not on my hands.  This is a MWE:  \documentclass{book}  \usepackage{titletoc} \titlecontents*{subsection}[5.4em]{\small} {\thecontentslabel~ }{}{}[.~\textbullet\ ][.]  \begin{document}  \tableofcontents  \chapter{Test Chapter 1} \section{Test Section 1.1} \subsection{First Test Subsection} \subsection{Second Test Subsection} \subsection{Third Test Subsection} \subsection{Fourth Test Subsection} \section{Test Section 1.2} \subsection{First Test Subsection} \subsection{Second Test Subsection}  \chapter{Test Chapter 2} \section{Test Section 2.1} \subsection{First Test Subsection} \subsection{Second Test Subsection} \subsection{Third Test Subsection} \subsection{Fourth Test Subsection} \section{Test Section 2.2} \subsection{First Test Subsection} \subsection{Second Test Subsection}  \end{document} Why does a subpanel need separate ground and neutral? Why does a power sub panel have to have separate neutral and ground bars other than code satisfaction?  Does a sub panel need a main breaker if there is one at the main?   Also, does the sub panel need a ground wire connected back to the main panel in the same building? When Bottom Navigation Isn't Best I'm developing a mobile app and dealing with various UI/UX issues. I'm open to all kinds of ideas at this stage, however some designers I consult with seem to be zealots for bottom-navigation and are hesitant in actually analyzing whether the bottom-nav is actually best for my app; they just want me to use the bottom tabs navigation as if it's some UI law.  One point I try to bring up is that my business model has users who aren't typical one-handed mobile users, are tech savvy, would use my app more of a tool, and would spend only a few minutes in it. The app isn't a typical lazy-content-browsing experience where you want to minimize fingers-over-top-of-screen interference, and where you want to keep the use transfixed and spending time in the app.  Anyways, I'd like to hear about situations where a bottom nav was decided-on, or started-with, and you had good reason to eliminate it and go with another navigation design.  Most articles out there talk about eliminating hamburger menus and diving into bottom nav. Is the converse simply going back to hamburger menu? What happens when a photon hits a mirror? When a photon of light hits a mirror does the exact same photon of light bounce back or is it absorbed then one with the same properties emitted? If the same one is bounced back does it's velocity take all values on $[-c,c]$ or does it just jump from $c$ to $-c$ when it hits the mirror?  Or, is the phenomenon of a mirror better explained using a wave analogy? If so, what is this explanation? Given that $$f\left(\frac{x+y}{2}\right)\leqslant \frac{f(x)+f(y)}{2}~,$$ how can I show that $f$ is convex. Thanks.      Edit: I'm sorry for all the confusion. $f$ is assumed to be continuous on an interval $(a,b)$.