For the equation $a^2 + b^2 = c^2$, the solution is:   $a = m^2 - n^2, b= 2mn, c = m^2 + n^2$ $m,n\in\mathbb{Z}$ and $m &gt; n$, free to choose   How is a similar solution obtained for the equation $a^2 + b^2 = c^2 + 1$?  I'm familiar with it, but don't know how to obtain it.
Are there integral solutions to the equation $x^2+y^2-z^2=1$?
How to install latest Code::Blocks from tar.xz? Does the everywhere differentiability of $f$ imply it is absolutely continuous on a compact interval? Suppose $f$ is differentiable everywhere on $[0,1]$. Must $f$ be absolutely continuous on $[0,1]$? I know this is true if $f'$ is integrable but I'm not sure in this more general case. Bind application to a specific network interface I have tried ForceBindIP, but it has a significant downside - it doesn't affect the children of the application I'm trying to bind, it only affects the application itself. It also can't force an application to always run through a specified interface, it has to be run through forcebindip.exe every time. It becomes a problem with applications like League of Legends where the process tree looks like that:    The launcher runs the patcher, the patcher run the client, etc. I can only affect the parent of all these processes in the tree, so the actual game is not bound to the interface I want, making this whole venture pointless.  Is there a more modern alternative to ForceBindIP for Windows 7? There are many questions similar to this one on this site, but they are mostly old. Maybe there is now a better way to solve this problem?  My current idea is to do the following:   Set up local 3proxy server bound to the desired interface. Run the game through Proxifier or similar software configured to run through that local proxy.   I'm not sure if that will work, but even if it will, it seems like a sub-optimal solution. Do you guys have any better ideas?  Edit: My idea didn't work :(  Edit 2: Basically, what I'm trying to achieve is bind a few applications to a regular interface, while VPN is running. The reason is that I need to connect through VPN most of the time, but some applications (such as games) don't work properly this way, because of higher ping and other issues. What is the correct pronunciation of the word "solder"? I have been listening to a podcast where the host pronounces the word solder as "sodder" or "sod-der", even "saw-der". Same thing happened when the lecturer of one of my EE classes pronounced the word solder. As for me I pronounce the word solder as "sole-der" with a distinct L and a longish O. I'm using X11 forwarding over ssh to run Linux apps on my Windows box, and when the network drops it loses everything that was running. Is there anything similar to screen for X11? I get this message while trying to install Ubuntu 12.04 as a guest in VirutalBox 4.1.18, on an Ubuntu 10.04 host.         This kernel requires the following feature not present on the CPU: pae   Some host specs:     The host's kernel is: Linux 2.6.32-41-generic-pae GNU/Linux   lscpu (host): Architecture: i686, CPU op-mode(s): 32-bit, 64-bit   grep --color=always -i PAE /proc/cpuinfo   does show pae in its output.   The 12.04 iso used is: ubuntu-12.04.0-desktop-i386.iso      As a comparison/check, I downloaded and installed Linux Mint 13 Cinnamon to the same host on the exact same VM (I just changed the .iso image). It worked fine. Its iso is: linuxmint-13-cinnamon-dvd-32bit.iso   It seems (to me) that I have pae.. what is going on here?   Update: I had assumed that Linux Mint also required pae (being Ubuntu based), but I've just run;   grep --color=always -i PAE /proc/cpuinfo   in the Mint VM.   It showed no output.   So it seems the issue may lie with VirtualBox.   If that is the case, how can I get Virtualbox into pae mode? A complete residue system from an arithmetic progression I'm solving a problem, and the solution makes the following statement: "The common difference of the arithmetic sequence 106, 116, 126, ..., 996 is relatively prime to 3. Therefore, given any three consecutive terms, exactly one of them is divisible by 3." Why is this statement true? Where does it come from? Is it generalizable to other numbers? Do we ever get more information about the Entwives? During my recent re-reading of The Two Towers I was struck again by a question that has bothered me since I was a youngling: What on earth is up with the Entwives?  Treebeard says they went away in the distant past and just... got lost?  Did Tolkien ever make mention of them again, or is this one of those unsolved mysteries of Middle Earth? How did Ron speak Parseltongue and re-enter the Chamber of Secrets? Although 'The Deathly Hallows' was overall an excellent conclusion to Harry Potter's saga, there were moments within the book that felt infuriatingly lazy, like Rowling was whipping the plot along regardless of the holes she tore in it. For me, the lowest point was when Ron gargles at the entrance of the Chamber of Secrets- and succeeds!- so he can get some Basilisk fangs to destroy a Horcrux (which are conveniently vulnerable to this venom).  This is, by itself, a shockingly slapdash deus ex machina, but even worse is how it is completely inconsistent with the guidelines Rowling herself set. First and foremost, Parseltongue is not just a language, but a magical ability that can only be learned through hereditary or magical means (Potter being the sole example of the latter). This is firmly established by the fact that even Dumbledore, the greatest wizard of all time and a polyglot, is completely unable to speak it despite understanding it perfectly.  Without linguistic knowledge nor inherent ability, the chances of making any kind of coherent sentence (let alone the right one!) would be incredibly slim even in the case of mundane languages; it would be like gibbering at a Filipino in the hope that what you said was fluent Tagalog and didn't insult their mother.  Other baffling issues somewhat related to my question are thus; even if Ron had somehow unlocked the gate, how did he make his way down? In the second novel, Lockheart's foolery demolished the main tunnel to the Chamber, making it impossible to access or leave; the fact that the teeth are still there also implies that the Chamber wasn't excavated and the rubble cleared in the interval. Granted, he could have tried Apparating, but then he wouldn't have needed to fake Parseltongue to begin with. Would you be weightless at the center of the Earth? If you could travel to the center of the Earth (or any planet), would you be weightless there? Is the Discrete Topology on $X$ the Only One Containing All Infinite Subsets of $X$? Prove or find counterexamples.  Let $X$ be an infinite set and $T$ be a topology on $X$. If $T$ contains every infinite subset of $X$, then $T$ is the discrete topology. The "I Get It" Reputation Problem I wrote a question about blocking certain characters in passwords. It became tremendously unpopular, and it seemed like everyone wanted to rain on my parade every day.  Eventually, I "got it," agreed completely with what people were saying to me, and just wanted to withdraw my question so that my reputation didn't slide even further downhill.  Unfortunately, the system is designed such that I can't withdraw my question or redeem myself. Instead, I just get daily reputation attacks. That's a terrible system, and it needs to be changed. Why am I getting "line 1: $' :\r': command not found"? I use Cygwin on my laptop (DOS). I have a collection of scripts from my colleagues, and my own. I am not an IT person, not knowledgeable in Unix. I am following my colleagues' syntax and able to manage a few simple things. The scripts worked well on my old laptop. I just changed laptop and installed Cygwin. When I run my scripts, they do not work. Here is one example of the error message I get: line 1: $':\r': command not found line 5: syntax error near unexpected token `$'\r'' line 5: `fi  Here are the top 5 lines of my script : iter=1 if [ -f iter.txt ]    then rm ./iter.txt  fi    Can someone please explain how I can get around this problem? Creating fillable PDFs I'd like to create a LaTeX document that when rendered into PDF, has forms that can be filled out using Adobe Reader or other such programs. Then I'd like to be able to extract the data. I deliberately would like to avoid using Acrobat for all the usual reasons (non-free, need different versions for different platforms etc).   Can this be done ? Bullied boy given immense powers by an alien race who destroys the human race Human-readable filesize with find's printf How to change the forgotten root password? I forgot my root password...actually I think I know what it is because it is the same password I have for pretty much everything but in the Terminal in tell me "Sorry, try again." Is there a way to reset the password? If so how? Mutable vs immutable objects I'm trying to get my head around mutable vs immutable objects. Using mutable objects gets a lot of bad press (e.g. returning an array of strings from a method) but I'm having trouble understanding what the negative impacts are of this. What are the best practices around using mutable objects? Should you avoid them whenever possible? Chrome: Site has updated in the background I keep getting these notifications despite not having an RT tab open. What gives? Group posts by year, then by month