Prove that $y^2z^2 - y^2 -z^2$ is not a perfect square for any $y,z \in \mathbf{N}$
Find all integer solutions to $x^2+y^2+z^2=2xyz$
How do I fix the GPG error "NO_PUBKEY"? Unable to install any updates through update manager/apt-get upgrade I have been running Ubuntu 12.04 for the past few weeks. I recently tried to download and install some files through the Update Manager. It located a few files, downloaded them, but then failed to install them. It gave the error:  installArchives() failed: (Reading database ...  (Reading database ... 80% dpkg: unrecoverable fatal error, aborting:  reading files list for package 'libasn1-8-heimdal': Input/output error   I tried using apt-get upgrade but a similar error was produced:  Reading package lists... Done Building dependency tree        Reading state information... Done The following packages will be upgraded:   icedtea-6-jre-cacao icedtea-6-jre-jamvm openjdk-6-jre openjdk-6-jre-headless   openjdk-6-jre-lib 5 upgraded, 0 newly installed, 0 to remove and 0 not upgraded. Need to get 0 B/34.6 MB of archives. After this operation, 1,024 B of additional disk space will be used. Do you want to continue [Y/n]? y (Reading database ... 80% dpkg: unrecoverable fatal error, aborting:  reading files list for package 'libasn1-8-heimdal': Input/output error E: Sub-process /usr/bin/dpkg returned an error code (2)   Any help would be greatly appreciated. Thanks in advance.  Edit: Here's some extra information that may be useful. My original problem was that the Ubuntu Software Centre would not open. I found that the Update Manager and Synaptic Package Manager also failed to open. They all gave an error along the lines of:   "installArchives() failed: dpkg: error: reading package info file '/var/lib/dpkg/available': Input/output errorError in function:"   After much googling I seemed to fix that problem as the package managers would open again. When I then tried to download and install any updates from the Update Manager I got the error at the beginning of this post. Why are these estimates to the German tank problem different? Suppose that I observe $k=4$ tanks with serial numbers $2,6,7,14$.   What is the best estimate for the total number of tanks $n$?   I assume the observations are drawn from a discrete uniform distribution with the interval $[1,n]$. I know that for a $[0,1]$ interval the expected maximum draw $m$ for $k$ draws is  $1 - (1/(1+k))$. So I estimate $\frac {k}{k+1}$$(n-1)≈$ $m$, rearranged so $n≈$ $\frac {k+1 }{k}$$m+1$.  But the frequentist estimate from  is defined as:  $n ≈ m-1 + $$\frac {m}{k}$  I suspect there is some flaw in the way I have extrapolated from one interval to another, but I would welcome an explanation of why I have gone wrong! How do I bypass restrictions on the length of the PATH variable How do I remap the keys on my keyboard? I broke my laptop and had to buy a new one with a new keyboard that I am struggling to get familiar with, especially with the keys like home, page-up, page-down and end, which aren't standardized at all.  I found myself doing this for like the 5th time in the last several years, and I think it is really annoying.   Is there an app that can reliably map one key to another?  By reliably I mean that key mapping works with Shift, Alt, and other modifiers.  UPDATE  Windows 7, Samsung Chronos 7 Springer llncs TOC not working I can't get the TOC to work. I remember it worked before, but there was something I did (unfortunately I can't remember what it was), and after that the TOC just stopped working.  Here is a part from my document:  \documentclass[deutsch]{llncs}  \usepackage{graphicx} \usepackage[labelformat=empty]{caption} \usepackage{subcaption} \captionsetup{compatibility=false}  \bibliographystyle{unsrt}  \pagestyle{headings} \begin{document}  \sffamily  \begin{titlepage}     %Some stuff here \end{titlepage}  \thispagestyle{headings} \tableofcontents \newpage %more stuff here blablabla \section{test} \subsection{test2} \end{document} In "", Mark Lutz mentions "mixins". I'm from a C/C++/C# background and I have not heard the term before. What is a mixin?   Reading between the lines of  (which I've linked to because it's quite long), I'm presuming it's a case of using multiple inheritance to extend a class as opposed to 'proper' subclassing. Is this right?   Why would I want to do that rather than put the new functionality into a subclass? For that matter, why would a mixin/multiple inheritance approach be better than using composition?  What separates a mixin from multiple inheritance? Is it just a matter of semantics? How do you transpose tensors? We transpose a matrix $A$ by replacing $A_{ij}$ with $A_{ji}$, for all $i$ and $j$.  However, in case $A$ has more than two dimensions (that is, it is a tensor), I don't know how to apply the transpose operation.   If A has dimensions $3\times 3 \times 8$, then what will replace $A_{ijk}$?  If $A$ has shape $3\times 3\times 8\times 8$, then what will replace $A_{ijkl}$? What does it mean to say "a divides b" I am not a number theorist and I am learning about relations.  I encountered a relation that says      $a \leq b$ if $a$ divides $b$   Can someone clarify what it means to a number to divide another number?  Does it mean what I think? $a$ divides $b$ if $a | b \in \mathbb{Z}$?     So given a set $S = \{a, a^2, a^3, \ldots\}$, with relation $a | b \leftrightarrow a \leq b$, does the relation hold   going from left to right or right to left? i.e. $a|a^2, a^2|a^3, \ldots$ $S_3$ permutes the elements of $\mathbb{R}^3$, show that $\sigma (x_1, x_2, x_3) = ( x_{\sigma^{-1}(1)}, x_{\sigma^{-1}(2)}, x_{\sigma^{-1}(3)})$. Both DHCP and static IP addresses simultaneously on one interface We can use multiple IP addresses for a single interface in Linux (eg. eth0:0, eth0:1). But can we use DHCP and static IP at the same time? eth0:1 should have a static IP address and eth0:0 should be configured using DHCP.  How can we accomplish both? When is 6 a square in Zp To find when $6$ is a square I divided $6$ in to $2$ and $3$ by The Legendre symbol.  $$\frac{6}{p} = \frac{2}{p} \times \frac{3}{p}$$  I know that we can multiply the cases of when $2$ and $$ are squares because then it is (1)(1) by Legendre.   $2$ is a square when $p \equiv 1 \pmod{8}$   $3$ is a square when $p \equiv 1 \pmod{12}$  so $6$ is a square when $p \equiv 1 \pmod {24}$ ($96$, but the $lcm(8,12)= 24$)   but how do I find the cases when $2$ and $3$ are both NOT squares? because then $6$ would be a square then as well, $(-1)(-1) = 1$ by Legendre. Can Hooke's law be derived? Can we derive  from the theory of elasticity? I know it is not a fundamental law and therefore can be derived from more basic considerations. I've got the following two tables (in MySQL): Phone_book +----+------+--------------+ | id | name | phone_number | +----+------+--------------+ | 1  | John | 111111111111 | +----+------+--------------+ | 2  | Jane | 222222222222 | +----+------+--------------+  Call +----+------+--------------+ | id | date | phone_number | +----+------+--------------+ | 1  | 0945 | 111111111111 | +----+------+--------------+ | 2  | 0950 | 222222222222 | +----+------+--------------+ | 3  | 1045 | 333333333333 | +----+------+--------------+  How do I find out which calls were made by people whose phone_number is not in the Phone_book? The desired output would be: Call +----+------+--------------+ | id | date | phone_number | +----+------+--------------+ | 3  | 1045 | 333333333333 | +----+------+--------------+ Any option for mysqldump to ignore databases for backup? Addition with NULL values I would like to ask a question not about software architecture, but physical architecture, i.e., the design of buildings and other elements of the "built environment".  I was looking for a Stack Exchange site to ask what the meaning is of the huge stone symbols on the side of the new Scottish Parliament building. 14.04 LTS No Dash / Launcher after update through Software Updater Let $G$ pe a $p$ group. I have to show that         the number of nonnormal subgroups is divisible by $p$       the number of subgroups differs from the number of normal subgroups by a power of $p$.   Are there any theorem that can help me to prove this ? We have discussed the Sylow theorems but I don't know how to apply them - if those are the theorems I need. (Does this theorem also hold for infinite groups $G$ ?) How do I copy to the clipboard in JavaScript?