Prove that there exist infinite positive integers n such that n and n + 1 are filled with squares in its factorization
Inductive proof that there are infinitely many pairs of consecutive powerful integers?
How to jusify {e,n} is a valid RSA public key? Do there exist an infinite number of 'rational' points in the equilateral triangle $ABC$? Let's call a point $P$ which satisfies the following condition 'a rational point'.  Condition: Each distance $PA, PB, PC$ from a point $P$ to three vertices $A, B, C$ of an equilateral triangle $ABC$ which has edge-length $1$ is rational number.  I'm interested in this point because I found the following three:  1. There exist an infinite number of rational points on an edge of the equilateral triangle $ABC$.  2. There exist an infinite number of rational points on the circumference of the circumscribed circle of the equilateral triangle $ABC$.  3. There exists a rational point in the equilateral triangle $ABC$.  Supposing that $A(0,\frac{\sqrt3}{2}), B(-\frac12,0), C(\frac12,0)$, one example for the above 3 is the following: $$P\left(\frac{61}{2058},\frac{220}{1029}\sqrt3\right), \left(PA, PB, PC\right)=\left(\frac{73}{147}, \frac{95}{147}, \frac{88}{147}\right).$$  I found this example by using computer. I found the other examples, but I don't know there exist an infinite number of rational points in the equilateral triangle $ABC$. Then, here are my questions.  Question 1: Do there exist an infinite number of rational points in the equilateral triangle $ABC$ ?  Question 2: Can we find all rational points in the equilateral triangle $ABC$ ? Order of finite fields is $p^n$ A few weeks back I was using std::ifstream to read in some files and it was failing immediately on open because the file was larger than 4GB. At the time I couldnt find a decent answer as to why it was limited to 32 bit files sizes, so I wrote my own using native OS API.  So, my question then: Is there a way to handle files greater than 4GB in size using std::ifstream/std::ostream (IE: standard c++)  EDIT: Using the STL implementation from the VC 9 compiler (Visual Studio 2008).  EDIT2: Surely there has to be standard way to support file sizes larger than 4GB. Is it possible to test apex that relies on field history tracking? Google Spreadsheets: IF not working I have problems with my IF command in Google Sheets. If gives #ERROR! all the time. See this simple test:    Not even if I put in the number directly and avoid references:    Am I using the IF wrong? Or what could such a problem be? This example is from a new document. Showing that Y has a uniform distribution if Y=F(X) where F is the cdf of continuous X Let $X$ be a random variable with a continuous and strictly increasing c.d.f. function $F$ (so that the quantile function $F^{−1}$ is well-deﬁned). Deﬁne a new random variable $Y$ by $Y = F(X)$. Show that $Y$ has a uniform distribution on the interval $[0, 1]$.  My initial thought is that $Y$ is distributed on the interval $[0,1]$ because this is the range of $F$.  But how do you show that it is uniform? How to find the sum of this series?  $$\sum_{k=0}^{\infty}\cfrac{{2}^{k}}{\binom{2k+1}{k}}$$  It seems very easy. But I still can not work it out, can anyone help? (p.236)  has a proof that the equation $e^z-z=0$ has infinitely many complex solutions:    I'm curious if there are any particularly elegant ways to see this, other than that given in the text. House wiring and grounding. How do you retrofit an old house and ground it? A contractor looked at our house and told me that you can't fix a house (with old wiring) that has some outlets that are not grounded.  He said that you can go on YouTube and search the internet and it will tell you that you can.  He says that the only way to properly fix and ground everything is to do a whole house rewire.  Is he correct?  If, theoretically, you have an old house that has no ground whatsoever, can you run a grounding wire from the panel to all of the outlets to fix this?  updated  We have BX/AC wiring. And also apparently we have "...old-style AC (BX) with cloth insulated wires in a paper overall wrap under the spiral armor (no bonding tape)" Courtesy of @ThreePhaseEel. In "Tasksel" how do I select a package? I found a video on YouTube that shows someone using the command tasksel from the server's boot menu after a fresh install of Ubuntu Server. Simple enough it looked like, but when I go to the options I want to choose for installation it has no information on how to select them, and when I click Enter on one of them it takes me back to the command prompt.  So how do I make a selection of the items listed under tasksel?  I have tried Shift++ to add, but no luck. Anyone know?  I want to install Ubuntu Desktop, as Ubuntu GNOME GUI is not an option in the listing for Ubuntu 17.10 Server. Let $F:\mathbb{R^2}\rightarrow \mathbb{R}$ be a function, such as : $$F(x,y)=\begin{cases}        0 &amp; (x,y)=(0,0) \\       \frac{xy}{x^2+y^2} &amp; (x,y)\neq (0,0)    \end{cases}$$      Is $F$ differentiable at $(0,0)$?   My attempt :  Consider the limit, $\lim_{(x,y)\rightarrow (0,0)} F(x,y)$ along the line $y=x$. The limit equals to $\frac{1}{2}$. However, $F(0,0)=0$. Hence $F$ is not continuous at $(0,0)$ which implies that $F$ isn't differentiable as well.  Is this approach okay ? Deep Format of a Harddrive on a Linux System I've got an original Asus Eee PC Netbook that I'd like to sell.  When I've sold old computers in the past, I've typically done a deep/secure/zeroing out style format.  How can I do this on a Linux Netbook?  I'm looking for   The command that will let me do this Advice on how I can do this on a device with (seemingly?) non-removable media   Thanks in advance! LaTeX package for syntax highlighting of code in various languages I am looking for a LaTeX package that does syntax highlighting on code. For example, right now I use the verbatim block to write code:  \begin{verbatim}     &lt;html&gt;        &lt;head&gt;            &lt;title&gt;Hello&lt;/title&gt;        &lt;/head&gt;        &lt;body&gt;Hello&lt;/body&gt;     &lt;/html&gt; \end{verbatim}   And this works fine to display the code on my document. But, suppose I wanted to highlight the HTML markup the way an IDE would in the output document? Is there a package that could help?  I would like to do the same for various languages such as Java, C#, HTML, CSS and so on. Sobolev inequality If $f\in H^2(\mathbb R^2)$, I want to show that    $||f||_{L^\infty}\le c||f||_{H^2}$ $||f||_{L^\infty}\le c||f||_{H^1} [1+\ln(1+||f||_{H^2})]$   For 1, I use $||f||_{L^\infty}\le \sup_{x\in \mathbb R^2} |f|\le\int_{\mathbb R^2} |\hat f(\xi)|d\xi = \int(1+|\xi|^2)^{-1}(1+|\xi|^2)|\hat f(\xi)|d\xi \le (\int (1+|\xi|^2)^{-2}d\xi)^2(\text {which is integrable in this case})||f||_{H^2}\to \text{is this correct?}$   But for 2, I am stucked. How can I get the "ln"? and how can I make it into a product of $H^1$ and $H^2$ norm? Is $\mathbf{Q}(\sqrt{2}, \sqrt{3}) = \mathbf{Q}(\sqrt{2}+\sqrt{3})$? Is $\mathbf{Q}(\sqrt{2}, \sqrt{3}) = \mathbf{Q}(\sqrt{2}+\sqrt{3})$ ?    $$\mathbf{Q}(\sqrt{2},\sqrt{3})=\{a+b\sqrt{2}+c\sqrt{3}+d\sqrt{6} \mid a,b,c,d\in\mathbf{Q}\}$$   $$\mathbf{Q}(\sqrt{2}+\sqrt{3}) = \lbrace a+b(\sqrt{2}+\sqrt{3}) \mid a,b \in \mathbf{Q} \rbrace  $$  So if an element is in $\mathbf Q(\sqrt{2},\sqrt{3})$, then it is in $\mathbf{Q}(\sqrt{2}+\sqrt{3})$, because $\sqrt{6} = \sqrt{2}\sqrt{3}$.  How to conclude from there? Let $X$ be a topological space. The diagonal of $X \times X$ is the subset $$D = \{(x,x)\in X\times X\mid x \in X\}.$$   Show that $X$ is Hausdorff if and only if $D$ is closed in $X \times X$.   First, I tried to show that $X \times X \setminus D$ is open using the fact that $X \times X$ is Hausdorff (because $X$ is Hausdorff), but I couldn't find an open set that contains a point outside $D$ and is disjoint to it... How to Upgrade Ubuntu 12.04.2, 12.04.3 to 12.04.4, the next Hardware Enablement Stack? I am currently using Ubuntu 12.04.2 32bit. I installed it using LiveCD.  Tomorrow, 23rd August 2013,Ubuntu 12.04.3 is released. I want to upgrade from Ubuntu 12.04.2 to Ubuntu 12.04.3 without using any LiveCD. Is it possible? If so please suggest me how can I do.  Actually while using Ubuntu 12.04.2 I have troubled with graphics. My graphics card is Intel i845G 64 MB. When Ubuntu starts it works fine and smooth without any lagging, but after sometime it hangs for few seconds (1 or 2 seconds) with a garbage screen and becomes sluggish. All windows and browsers start lagging and also it is not possible to play any video in any player (VLC, Movie Player, Xnoise, SMPlayer etc..).  I think Upgrading to Ubuntu 12.04.3 could fix my problem.  Any help will be greatly appreciated..    I've posted an answer to upgrade Ubuntu 12.04.x to 12.04.4. I see the following error while trying run the command shown below. I read somewhere that my /boot partition is low on disk space. How can I increase the size of the /boot partition so I can install more software? I have a 500GB hard disk, so there is enough space to play with.  sudo apt-get install libdvdread4  gzip: stdout: No space left on device                                                                                                                                                                                  E: mkinitramfs failure cpio 141 gzip 1                                                                                                                                                                                 update-initramfs: failed for /boot/initrd.img-3.2.0-33-generic with 1.                                                                                                                                                 run-parts: /etc/kernel/postinst.d/initramfs-tools exited with return code 1                                                                                                                                            Failed to process /etc/kernel/postinst.d at /var/lib/dpkg/info/linux-image-3.2.0-33-generic.postinst line 1010.                                                                                                        dpkg: error processing linux-image-3.2.0-33-generic (--configure):                                                                                                                                                      subprocess installed post-installation script returned error exit status 2                                                                                                                                            dpkg: dependency problems prevent configuration of linux-image-server:                                                                                                                                                 linux-image-server depends on linux-image-3.2.0-33-generic; however:                                                                                                                                                    Package linux-image-3.2.0-33-generic is not configured yet.    dpkg: error processing linux-image-server (--configure):     dependency problems - leaving unconfigured    dpkg: dependency problems prevent configuration of linux-server:     linux-server depends on linux-image-server (= 3.2.0.33.36); however:      Package linux-image-server is not configured yet.    dpkg: error processing linux-server (--configure):     dependency problems - leaving unconfigured    No apport report written because the error message indicates its a followup error from a   previous failure.                                                                                                           No apport report written because the error message indicates its a followup error from a previous failure.  Errors were encountered while processing:  linux-image-3.2.0-33-generic  linux-image-server  linux-server N: Ignoring file 'michael-gruz-canon-precise.list.1' in directory '/etc/apt/sources.list.d/' as it has an invalid filename extension N: Ignoring file 'michael-gruz-canon-precise.list.1' in directory '/etc/apt/sources.list.d/' as it has an invalid filename extension   Listed below is the output of du  Filesystem              1K-blocks      Used Available Use% Mounted on /dev/mapper/ubuntu-root 712660664 104095912 572363692  16% / udev                      3964792         4   3964788   1% /dev tmpfs                     1591012      1064   1589948   1% /run none                         5120         0      5120   0% /run/lock none                      3977528       684   3976844   1% /run/shm /dev/sda1                  233191    219821       929 100% /boot How to make a complete factory reset, without anyone being able to retrieve my data? I will have a few Android phones that I will have to let go.   I would like to make a proper clean wipe of the data, without no one being able to retrieve them, and do a factory reset.  I have read about the , encrypting the device before doing a factory reset, doing both, etc.  Since I am not an Android/adb guru, I would like to know if some of you would have some idea on how to achieve this with certainty.  I know "certainty" is a big word when it comes to proper data erasure on flash memory but still.  I have found some answers here and somewhere else, saying that encrypting and doing a factory reset on your Android device is enough, but I could not find a reliable source.  Just a few information that could be useful:   I'm running Linux as my OS and, therefore, I'm not looking for a Windows-based solution My devices are non-rooted and (almost) identical (Samsung A510F + 1 A510FD)