Prove that $x^2 + y^2 = z^n$ has a solution in  $\mathbb{N}$, for all $n$ belonging to the set of natural numbers.    I think it uses Induction as a method of proof but just can't proceed with three variables. Any help is greatly appreciated.
I was presented with the following problem:      Prove that there exist solutions to $x^2+y^2=z^n$ for all $n$, with $x,y,z, n \in \mathbb{N}$   I showed that by taking any Pythagorean triple $x^2+y^2=z^2$ and multiplying by $z^{2(n-1)}$ we get $(z^{n-1}x)^2+(z^{n-1}y)^2=(z^2)^n$, which allows me to generate solutions easily for any value of $n$. I managed to find several similar questions on this site, such as  concerning the specific case $n=3$. I notice that all of these questions take a similar approach and start with a Pythagorean triple and use it to generate general solutions.  Is there a way to prove the statement (or better yet provide solutions to the equation) without first relying on Pythagoras?
How do I install the latest version of EMACS and pick the appropriate UI? I'm running the latest Kubuntu (12.04 beta 2) and I would like to run the latest emacs (currently v24). The repos are one version behind. What's the best way to install v24 or later (and avoid future version conflicts)?  Also, is there any reason not to aways use the GUI version of emacs if X is running? For example, could I set the GUI emacs version as the default text editor and use it to edit cron jobs (crontab -e)? I'm assuming the answer is yes, but since I haven't done that yet (my default editor is nano), I want to check if there are reasons I should leave nano as the default editor.  Usually when I'm working on the command line I end up using nano. Now that I think about it, I have no idea why I keep doing that. Is there any downside to calling a GUI editor when working in an X terminal?  EDIT: I briefly tested these two versions   GNU Emacs 24.0.94.1 (x86_64-pc-linux-gnu, GTK+ Version 3.3.20) from  GNU Emacs 23.3.1 (x86_64-pc-linux-gnu) installed by default in Kubuntu.   .  Unfortunately (for me) the defaults installed version (23.3.1, 23.3+1-1ubuntu9) is the nox version.   Package: emacs23-nox Status: install ok installed Version: 23.3+1-1ubuntu9 Replaces: emacs23, emacs23-gtk, emacs23-lucid   The package with version 24 opens in GUI mode by default. That's what I prefer. Some of the version 24 changes that interest me are listed in the references below.   But there appear to be a multitude of different packages and versions I could install.  References:   What’s New In Emacs 24 (part 1) | Mastering Emacs  "    shell-mode uses pcomplete rules, with the standard completion UI.  Yowzah! There’s a lot of cool, new functionality hidden away in this gem of a change." EmacsWiki: Recent Changes Make an exam format with a logo Why doesn't the last digit method work for divisibility by 4? I wish to explain to a grade 5 kid, the answer to the question - 'When can we know that the observing the last digit of a number will not be sufficient to determine its divisibility by a given N?'  For example - It doesn't work for 3, because we get all sorts of digits, i.e. from 0 to 9, at the units place. The same reason works for 7 and 9.   But what would be a convincing argument for 4 and 8? Even though the last digits form a pattern - 0, 2, 4, 6 and 8 - but we're still not sure about its divisibility. What are the technical differences between using a DSLR viewfinder and live view? The earlier question  asks for practical advantages of optical viewfinders. But what's the technical difference, and why does a camera behave differently in each mode?  For example, on my Nikon D5100:   Slower auto-focus — I saw this explained in the related issue, but just partially: "The other major drawback of live view is that because it requires the mirror to be locked up, it means that the auto focusing mechanism is unavailable."  option not available. No . The manual says that it's dangerous to shoot direct light, when it's on live-view. There's no range finder available.   Why are all these things different?  Live view also has some advantages, like:   automatic scene selection grid lines on the screen other smaller things, like possibly easier operation on a tripod where the viewfinder might be inconvenient to look through   Some of these things, like the tripod operation, are natural. But are there technical reasons the others can't be done with an optical finder? Are there other advantages enabled by live view for technical reasons? How can I sort arrays and data in PHP? Using same modulus for RSA Web colors: how to compensate for differences in monitors Positioning relative to page in TikZ WinError 32 error when trying to delete shapefiles with QGIS python I am working on the following problem from group theory:     If $G$ is a group of order $2n$, show that the number of elements of $G$ of order $2$ is odd.   That is, for some integer $k$, there are $2k+1$ elements $a$ such that $a \in G,\;\; a*a = e$, where $e$ is the identity element of $G$. How do I calculate the resistor value for a simple LED circuit? Why do Ruby setters need "self." qualification within the class? lstlisting directly after section start removes margin before next section If I place an lstlisting directly after the start of a (sub)section, the margin before the following (sub)section (title) disappears.  Here is a an example document showing the issue:  \documentclass[twocolumn]{article} \usepackage{listings} \begin{document}  \section{Lorem ipsum} Consectetur adipiscing elit.  \begin{lstlisting} Lorem \end{lstlisting}  Sed quis ante nec mi volutpat congue eu id leo.  \section{Dolor sit amet} Suspendisse at lorem mattis, semper massa id, euismod risus.  \newpage  \section{Lorem ipsum} \begin{lstlisting} Lorem \end{lstlisting}  Sed quis ante nec mi volutpat congue eu id leo.  \section{Dolor sit amet} Suspendisse at lorem mattis, semper massa id, euismod risus.  \end{document}   And you can see the (annotated) result here:    Why is there no margin before the fourth section (title)? Is this a bug in my LaTeX environment (MacTeX-2013), or is this the expected behavior? How to show that there is an element of order $51$ in the multiplicative group $(Z/103Z)^∗.$  I can't use Cauchy's theorem since $51$ is not prime. Characterizing non-constant entire functions with modulus $1$ on the unit circle I host am ExpressionEngine based website on a single Amazon EC2 instance in the US East (Virginia) region. I use Ubuntu.    The website allows members to post their content, and displays it for other members and for users who are not logged in.   Goal: I need to begin serving the same website across the globe, equally fast to Hong Kong, New York, and Los Angeles.   I considered two solutions:  1.Setting up three separate EC2 instances for three regions: Virginia, Oregon, and Singapore. Use Amazon Rot53 Geo Location to direct traffic to instances based on the browser's IP address.   Pro: the website will load fast for the region.   Contra: these will be essentially three separate websites. Unless I am mistaken?   Setting up one instance in a region equally remote from all three locations: (Ireliand), and boost the server to the highest performance available.    Pro: It will be the same website for all regions.   Contra: it will cost a lot to make it fast enough.   My challenges:   a) I do not know a way to sync server settings, database, and content between multiple Ubuntu instances in real time. If a knowledgeable person could explain to me how to do that, I would be most grateful.   b) I am not thoroughly familiar with all Amazon AWS resources. Does AWS has a built-in method that would allow me to achieve my goal of serving a single PHP / MySQL based membership website across the globe equally fast in all three chosen locations? When scripting in bash or any other shell in *NIX, while running a command that will take more than a few seconds, a progress bar is needed.  For example, copying a big file, opening a big tar file.  What ways do you recommend to add progress bars to shell scripts? I use to work at night and it is eye-hurting to watch at lightbulb-bright screens (I've got two - laptop's built-in panel and an external 18-inch CRT) in a dark environment. So I adjust my screens to be darker.  Can I do this a software way instead of pressing monitors' buttons that many times every time? Student missing exam for “medical reasons” What is the geometric interpretation of the transpose?