For each prime $p$, let $ X_p=\{(a,b)\in\{1,...,p-1\}^2{\,{\large{\mid}}\,}a^p+b^p\equiv 1\;(\text{mod}\;p^2)\} $.  Based on limited data testing, the following conjectures appear to hold:  $(1)\;\;X_p\ne{\large{\varnothing}}\;$for all primes$\;p\equiv 1\;(\text{mod}\;6)$.  $(2)\;\;X_p={\large{\varnothing}}\;$for infinitely many primes$\;p\equiv -1\;(\text{mod}\;6)$.  Any thoughts about the validity of these claims?
Consider $X = (a+b)^n-a^n-b^n$  where  $a$ and $b$ and $a+b$ are not divisible by $n$, $n&gt;=3$ is positive and prime, $a \ne b$ and both are greater than 1.  By the binomial expansion we know $X$ is divisible by $n$, but how can we study whether $X$ is divisible by $n^2$ or other powers of $n$? Specifically I am interested to prove $X$ is NOT divisible by $n^2$, but I am not sure how to prove or disprove it.
WCF server-side timeout How exactly do I install Minecraft Mods and what is Forge? If I select a feature and run self.iface.actionIdentify().trigger() it will open the identify window but won't open the selected feature's data, how can I make it open the data of the selected feature? 14 months ago, an  supplement was published that discussed how the ranger described in the Player's Handbook was under powered. These rules reimagined the ranger's abilities, and suggested that an official version would be released later.     Finally, we come to implementation. If this iteration of the ranger, or a future revision of it, grades high enough, our plan is to present it as a revised ranger in a future D&amp;D sourcebook.   Does that Sourcebook exist yet? If not, have the devs given an update or told us their intentions? $3$ and $5 $cent coins How do I backup ssh keys? I am planning to do a fresh install of ubuntu 11.10 in my system. Before that i have setuped key based ssh authentication in this machine. Is it possible to take backup of those ssh keys, so that i  can use that in my new installation. Or else i must setup keybased ssh authentication again? If i can take backup, what are the files i need to copy? Can someone explain it in detail pls. Thanks in advance. I'm trying to include graphics in my Latex-file, which I compiled with latex+dvipdf on OS X. Latex however returns this error: "Cannot determine size of graphic"  My graphic is exported from PowerPoint, so I have tried both .pdf and .png. I get the same errors for both.  Sample code that doesn't work:  \begin{figure}[htb] \begin{center} \leavevmode \includegraphics[width=0.8\textwidth]{graph.png} \end{center} \end{figure}   If I have to manually set some measurements of the image, please tell me how I can find out those measurements. I need the image to take up about 80% of text width and be centered. Thanks. Proving $T$ and $U$ are equal if I know $(X,T)$ and $(X,U)$ are hausdorff and compact This question is from Wayne Patty Exercise 4.2  Let $T$ and $U$ are Topologies on set X. Suppose $ (X,T) $ and $(X,U)$ are compact Haurdsorff Spaces. Prove that $T=U$ or $T$ is not a subset of $U$ or $U$ is not a subset of T.  I thought of proving it by this way: Assuming that if $T$ is a subset of $U$ or $U$ is a subset of $ T$ then $T=U$ must be true. But I was unable to think on how to use compactness or hausdorff property and was unable to proceed. Well, also this question was asked before on this website. But I am unable to understand how the solution will work despite trying to understand it for a fair amount of time as it was just outlined. So, I am looking for a detailed solution. What are best practices for managing SSH keys in a team? I work with small teams (&lt;10) of developers and admins with the following characteristics:   Most members of the team have >1 personal computer, most of which are portable Team members have access to 10-50 servers, usually with sudo   I think this is pretty typical for most startups and small- to mid-size corporate IT groups.  What are the best practices for managing SSH keys in a team like this?  Should you share a single key for the whole team?  Should each person have their own key on a shared account ("ubuntu" on every server)?  Separate accounts?  Should each team member keep a separate key for each of their laptop or desktop computers? Granted there are already quite a few badges for community service on this site, I thought, how about some more!  The theme is merging enthusiast/fanatic with suffrage/vox populi. Something like this (names and numbers open to suggestions)  Enthusiastic Voter - Voted 20 times per day consecutively for 30 days ????? - Voted 30 times per day consecutively for 50 days Fanatic Vox - Use the maximum number of 40 votes consecutively for 100 days When I use the , do I choose the monster to distract (and then the next ally to attack gets advantage) or do I choose the ally and the monster (so that only his attack against that target gets advantage)? Or something else?  Help [...] Alternatively, you can aid a friendly creature in attacking a creature within 5 feet of you. You feint, distract the target, or in some other way team up to make your ally’s attack more effective. If your ally attacks the target before your next turn, the first attack roll is made with advantage. Understanding NSString comparison iOS: How to change app language programmatically WITHOUT restarting the app? I have this alias set in my system /etc/bashrc file:  alias root="sudo !!"   The intention of this being to run the last used command using sudo of course. When used, it of course appears to substitute the last command in history to the bashrc file upon shell initialization, and not the actual command that you would get if you were to run sudo !! in an interactive shell. I've also tried alias root="sudo fc -s" to no avail.  I realize this is probably something to do with how the BASH does command substitutions, but can someone explain why this is, and provide a usable substitute?  I'm running BASH version 3.2.51(1)-release (x86_64-apple-darwin13). How do I convert between big-endian and little-endian values in C++? How do I convert between big-endian and little-endian values in C++? For clarity, I have to translate binary data (double-precision floating point values and 32-bit and 64-bit integers) from one CPU architecture to another.  This doesn't involve networking, so ntoh() and similar functions won't work here.  Note: The answer I accepted applies directly to compilers I'm targeting (which is why I chose it).  However, there are other very good, more portable answers here. I work in D8, I create a custom module first_module. By modules create a url and display all nodes by *.html.twig file.  Write code in controller.php  public function contentthree() {    $connection = Database::getConnection();    $query = $connection-&gt;select('node', 'n');   $query-&gt;fields('n', ['nid', 'type']);    $groups = $query-&gt;execute()-&gt;fetchAll();    foreach ($groups as $group) {     $events[] = [$group-&gt;nid, $group-&gt;type];   }    return [     '#theme'  =&gt; 'first_module_display',     '#events' =&gt; $events,   ]; }   In *.module  function first_module_theme($existing, $type, $theme, $path) {    return [     'first_module_display' =&gt; [       'variables' =&gt; ['events' =&gt; NULL],       'template'  =&gt; 'first-module-list',     ],   ]; }  function _get_node_details() {    $output = 'Welcome D8';    $output .= 'Node Details';    return $output; }   and create templates first-module-list.html.twig and display data  {%for key, data in events %}  {%for key1, data1 in data %}     {{ data1 }}  {% endfor %}  {% endfor %}   work fine and showing data correct. But, my question is how to call function _get_node_details() and how to render node details in first-module-list.html.twig files? Flag Options Missing - Is this a bug? This is normally what I see when flagging a post:   Which then provides the following options:    I have come across this problem before. Where I have wanted to flag a question, as not belonging here, but only have limited options.    When this happens, I use a custom flag and write in the reason, but am not happy doing so.  Is this a bug?   What is the best thing to do? Grub rescue - error: unknown filesystem I have a multiboot system set up. The system has three drives. Multiboot is configured with Windows XP, Windows 7, and Ubuntu - all on the first drive. I had a lot of unpartitioned space left on the drive and was reserving it for adding other OSes and for storing files there in the future.  One day I went ahead and downloaded Partition Wizard and created a logical NTFS partition from within Windows 7, still some unpartitioned space left over. Everything worked fine, until I rebooted the computer a few days later.  Now I'm getting:    error: unknown filesystem.   grub rescue&gt;   First of all I was surprised not to find any kind of help command, by trying:    help, ?, man, --help, -h, bash, cmd, etc.  Now I'm stuck with non-bootable system. I have started researching the issue and finding that people usually recommend to boot to a Live CD and fix the issue from there. Is there a way to fix this issue from within grub rescue without the need for Live CD?  UPDATE  By following the steps from , I was able to boot to initramfs prompt. But not anywhere further than that.  So far from reading the manual on , I was able to see my drives and partitions using ls command. For the first hard drive I see the following: (hd0) (hd0,msdos6) (hd0,msdos5) (hd0,msdos2) (hd0,msdos1)  I now know that (hd0,msdos6) contains Linux on it, since ls (hd0,msdos6)/ lists directories. Others will give "error: unknown filesystem."  UPDATE 2  After the following commands I am now getting to the boot menu and can boot into Windows 7 and Ubuntu, but upon reboot I have to repeat these steps.    ls ls (hd0,msdos6)/ set root=(hd0,msdos6) ls / set prefix=(hd0,msdos6)/boot/grub insmod /boot/grub/linux.mod normal   UPDATE 3  Thanks Shashank Singh, with your instructions I have simplified my steps to the following. I have learned from you that I can replace msdos6 with just a 6 and that I can just do insmod normal instead of insmod /boot/grub/linux.mod. Now I just need to figure out how to save this settings from within grub itself, without booting into any OS.  set root=(hd0,6) set prefix=(hd0,6)/boot/grub insmod normal normal   UPDATE 4  Well, it seems like it is a requirement to boot into Linux. After booting into Ubuntu I have performed the following steps described in the :  sudo update-grub sudo grub-install /dev/sda   This did not resolve the issue. I still get the grub rescue prompt. What do I need to do to permanently fix it?  I have also learned that drive numbers as in hd0 need to be translated to drive letters as in /dev/sda for some commands. hd1 would be sdb, hd2 would be sdc, and so on. Partitions listed in grub as (hd0,msdos6) would be translated to /dev/sda6.  UPDATE 5  I could not figure out why the following did not fix grub:  sudo update-grub sudo grub-install /dev/sda   So I downloaded  based on an answer from  post. That seemed to do the trick after I picked the "Recommended Repair (repairs most frequent problems)" option. Is there a rule in British English about how to pronounce "either"? Will the Windows 10 Free Upgrade stay in the Professional channel?