Wilson and Fermat's Theorem Question!     Show that the pair of positive integers $n$ and $n+2$ are twin primes if and only if $$4[(n-1)! + 1] + n = 0 \ (\text{mod} \ n(n+2)),$$ where $n$ does not equal $1$.
$(m, m+2)$ is twin prime, iff $4((m-1)! + 1) \equiv -m \pmod {m(m+2)}$ The Wiki page on  says     The pair $(m, m+2)$ is twin prime, iff $4((m-1)! + 1) \equiv -m \pmod {m(m+2)}$.   This is obviously connected to Wilson's Theorem. Can anybody provide a proof for that, along the lines of thought given here, ?
Desoldering/soldering 1/4" chip with eight pins I wonder if there is any difference between mapping and a function. Somebody told me that the only difference is that mapping can be from any set to any set, but function must be from $\mathbb R$ to $\mathbb R$. But I am not ok with this answer. I need a simple way to explain the differences between mapping and function to a lay man together with some illustration (if possible).  Thanks for any help. I am writing my thesis,  and i am in restructuring phase - sorting theorems where they should belong thematically. And it would be helpful to have all the proofs hidden so i am not laden with the additional information.  Is it possible to add something like hideproof to documentclass options to omit them when compiling the latex file. Suppose there are two users that have commented on an answer. Both have the same user name, and I want to reply to one of them. How can I do this?  It appears that, under the current system, there is no way for me to do this. Specifically, from , I see that:     4. Matching is performed in reverse chronological order, so if five people named John are participating, @john will match the most recent John   So, consider this a request to add this feature using an -n suffix or something similar; e.g. @john-2 could be used to reply to the second John from the top.  Update Thanks to  for nice answer of @name(userid) based auto-complete suggestion which has no loophole. It involves no counting from top &amp; has no consequences when a comment is deleted. But, I think @name(userid) should be used directly in comment as all form inputs at SE are text (nothing embedded).  Now, I'm trying to finalize it based on it: The system should allow optional @name(userid) format for reply. Provided userid with this format, system should ignore name. And, in the situation of duplicate name based confusion, system should demand this format as mandatory to submit comment.  How to get userid? In case of duplicate name based confusion, system should append (userid) in hypertext to their profile. It'd work for mobile version too. On full version, for easy fill-up, a hover reply icon would be fine (like hover vote up icon). Upon clicking it, system should fill @name(userid) in comment text box.  From reply reader side: Multiple Johns could be confused even if they didn't come from notification (because nobody remembers his/her own userid). So, highlight the background of comments which are replied to them. How do I restore /usr/bin/env after overwriting it with start-tor-browser file I was installing Tor and wanted to access it directly from the terminal, so I was trying to copy start-tor-browser to /usr/bin. But by mistake, I have replaced the /usr/bin/env file with the start-tor-browser file. What should I do now?? CSS - Make divs align horizontally I have a container div with a fixed width and height, with overflow: hidden.  I want a horizontal row of float: left divs within this container. Divs which are floated left will naturally push onto the 'line' below after they read the right bound of their parent. This will happen even if the height of the parent should not allow this. This is how this looks:  ![Wrong][1] - removed image shack image that had been replaced by an advert  How I would like it to look:  ![Right][2] - removed image shack image that had been replaced by an advert  Note: the effect I want can be achieved by using inline elements &amp; white-space: no-wrap (that is how I did it in the image shown). This, however, is no good to me (for reasons too lengthy to explain here), as the child divs need to be floated block level elements. "the better" vs "the best" of two players The uniform limit of Continuous Functions is continuous.   Proof Let $\varepsilon &gt; 0$. There exists $N$ in natural numbers such that $n&gt;N$ implies  $|f_n(x)-f(x)| &lt; \frac{\varepsilon}{3}$ for all $x$ in $S$. In particular $|f_{N+1}(x)-f(x)| &lt; \frac{\varepsilon}{3}$ for all $x$ in $S$. Since $f_{N+1}$ is continuous at $x_0$ there is a $\delta&gt;0$ such that, $x$ in $S$ and $|x-x_0| &lt; \delta$ imply $|f_{N+1}(x)-f_{N+1}(x_0)|&lt; \frac{\varepsilon}{3}$ Now we conclude $x$ in $S$ and $|x-x_0| &lt; \delta$ imply $|f(x)-f(x_0)|&lt; 3 \cdot \frac{\varepsilon}{3} = \varepsilon$.  Can someone explain what is happening here? I don't follow it all. Missing "References" heading How do I alter a message in the message queue independent of site language? I want to programatically alter a message in the message queue. The message I want to alter is set in the message queue by function comment_form_submit() in comment.module, i.e.:  if ($comment-&gt;status == COMMENT_NOT_PUBLISHED) {   if (!user_access('administer comments')) {     drupal_set_message(t('Your comment has been queued for review by site administrators and will be published after approval.'));   } }   Instead of this, I want the call to drupal_set_message() to be this:  drupal_set_message(t('Your comment will be published when you validate your e-mail address.'));   How do I do this?  Postscript: Based on the answer below by Topsitemakers, I got this working for any language by using array_search() inside a custom MYMODULE_comment_submit to search the message array, and then compare the translated strings to identify the key of the string to alter.  This is the working code I ended up with:  $mm = drupal_get_messages('status', TRUE); $key = array_search(t('Your comment has been queued for review by site administrators and will be published after approval.'), $mm['status']); if (FALSE !== $key) {   $mm['status'][$key] = t('Your comment will be published when you validate your e-mail address.'); } foreach ($mm['status'] as $msg) {   drupal_set_message($msg, 'status'); } "An error occurred when reviewing this item" On approving an edit, I just get this pop up since morning. At first I thought there might be some internet connection issue, but this is now the 3rd to 4th time I got the same error; also, the review is accepted because on refresh I can see my name with approved text beside, but it still throws this error.    I am on a Windows 7 machine, running Firefox 24 (Icognito/Private Mode)    Also, this happens only when edit(1) is written on an question/answer, I don't go to review queue and approve, these reviews were directly loaded from the notification beneath the question/answer.  Martijn Pieters reproduced the problem from the review queue three times; each time the improve option was picked, accepting the suggested edit but editing the post some more. The network console shows that posting to https://stackoverflow.com/review/task-reviewed fails, the server responds with a 404 error:  Request headers: POST /review/task-reviewed HTTP/1.1 Host: stackoverflow.com Connection: keep-alive Content-Length: 137 Accept: application/json, text/javascript, */*; q=0.01 Origin: http://stackoverflow.com X-Requested-With: XMLHttpRequest User-Agent: Mozilla/5.0 (Macintosh; Intel Mac OS X 10_7_5) AppleWebKit/537.36 (KHTML, like Gecko) Chrome/31.0.1650.63 Safari/537.36 Content-Type: application/x-www-form-urlencoded DNT: 1 Referer: https://stackoverflow.com/review/suggested-edits/3575396 Accept-Encoding: gzip,deflate,sdch Accept-Language: en,en-US;q=0.8,nl;q=0.6,nb;q=0.4,no;q=0.2,de;q=0.2,fr;q=0.2 Cookie: &lt;redacted&gt;  POST data: taskTypeId=1&amp;reviewTaskId=3575396&amp;taskResultTypeId=5&amp;fkey=f8791a329b445b658876f4d3afe7961e&amp;postCommentId=null&amp;taskResultSubtypeFlags=null  Response headers: HTTP/1.1 404 Not Found Cache-Control: private Pragma: no-cache Content-Type: application/json X-Frame-Options: SAMEORIGIN Date: Fri, 13 Dec 2013 12:04:53 GMT Content-Length: 0   On the  itself, no record is made of the accept; instead the suggested edit is recorded as approved by 3 people not including Martijn Pieters. In this case the improve review arrived too late.  In , improve again resulted in the error, but the Community account is shown rejecting the edit. Is the collection of all functions between two sets a set? Is there a way to "hide" a normal windowed application in Windows?  I'm looking for something that removes it from the taskbar and possibly puts an icon in the system tray instead. I have a bunch of client point of sale (POS) systems that periodically send new sales data to one centralized database, which stores the data into one big database for report generation.  The client POS is based on PHPPOS, and I have implemented a module that uses the standard XML-RPC library to send sales data to the service. The server system is built on CodeIgniter, and uses the XML-RPC and XML-RPCS libraries for the webservice component. Whenever I send a lot of sales data (as little as 50 rows from the sales table, and individual rows from sales_items pertaining to each item within the sale) I get the following error:  Fatal error: Allowed memory size of 134217728 bytes exhausted (tried to allocate 54 bytes)   128M is the default value in php.ini, but I assume that is a huge number to break. In fact, I have even tried setting this value to 1024M, and all it does is take a longer time to error out.  As for steps I've taken, I've tried disabling all processing on the server-side, and have rigged it to return a canned response regardless of the input. However, I believe the problem lies in the actual sending of the data. I've even tried disabling the maximum script execution time for PHP, and it still errors out. How to get horizontal black line when coloring cells? Why am I getting objects printed twice? Run multiple Python scripts in Blender from the command-line I have 8 python scripts and I need to lunch them at once. For one script, I used   blender --background --python script1.py   for the eight scripts, I tried this command but it does not work.  blender --background --python *.py Ever since I upgraded to 15.10, fdisk -l reports 16 ram disks (/dev/ram0 ... /dev/ram15). I'm a bit unsure what those are needed for. Is it safe to delete them? If not, how can I get rid of that fdisk output? Youtube video lag on 18.04 I just upgrade from Ubuntu 17.10 to Ubuntu 18.04, there is an issue when playing youtube video.The video stops play but audio playing, then I move the cursor &amp; video is playing...... What should I do?? Heisenberg Uncertainty Principle scientific proof states that:    $$\sigma(x)\sigma(  p_x )\ge \frac {\hbar}{2}.$$  What is the scientific proof of this principle?