How to prove that there are infinitely many integers n for which each of n, n+1, . . . ,n+ 1000 is composite.
Prove that $n! + k$ is a composite number
I encountered the following induction proof on a practice exam for calculus:  $$\sum_{k=1}^nk^2 = \frac{n(n+1)(2n+1)}{6}$$  I have to prove this statement with induction.  Can anyone please help me with this proof? Exit strategies for "chameleon questions" Are Destiny characters backwards and/or cross-compatible? From what I heard and remember hearing correctly, characters that are leveled up through the PS3 can be transferred to the PS4 version (assuming that both consoles have the game).  So what if I wanted to go back to the PS3 version with some friends? I only have the PS4 version at the moment--would my PS4 characters transfer over to the PS3 side? What if I wanted to go to the 360 version too to play with some other people too? Modules over commutative rings I'm trying to get my head around modules, and there's a problem that's bothering me regarding scalar multiplication from the left vs from the right.  In many books/articles I've read, the author may refer to a left $R$-module $M$, and then continue to talk of terms such as $xr$, where $x\in M$, $r\in R$. What is right scalar multiplication in a left module? Does it even make sense to talk about it?  I've been trying to show that over a commutative ring, right scalar multiplication and left scalar multiplication are the same thing, but I've not managed it so far. I've googled it to no avail, and I can't see why the distinction between left and right modules disappears when the base ring is commutative.  Thanks for any replies! I am using the below script to move two days back when script runs at starting two days of the year and also check first and second days of every month and move two days back. if [$month=&quot;01&quot;] &amp;&amp; [$day=&quot;01&quot;]; then         date=&quot;$last_month/$yes_day/$last_year&quot;         fulldate=&quot;$last_month/$yes_day/$last_year&quot; else         if [$month=&quot;01&quot;] &amp;&amp; [$day=&quot;02&quot;]; then                 date=&quot;$last_month/$yes_day/$last_year&quot;                 fulldate=&quot;$last_month/$yes_day/$last_year&quot;         else                 if [ $day = &quot;01&quot; ]; then                         date=&quot;$last_month/$yes_day/$year&quot;                         fulldate=&quot;$year$last_month$yes_day&quot;                 else                         if [ $day = &quot;02&quot; ]; then                                 date=&quot;$last_month/$yes_day/$year&quot;                                 fulldate=&quot;$year$last_month$yes_day&quot;                         else                                 date=&quot;$month/$yes_day/$year&quot;                                 fulldate=&quot;$year$month$yes_day&quot;                         fi                 fi         fi fi  But my bad am getting the below error message Etime_script.sh: line 19: [06=01]: command not found Etime_script.sh: line 24: [06=01]: command not found I want to remotely control my Android device from my PC without rooting it. Is this possible? Does 'rm .*' ever delete the parent directory? The expression .* is expanded by bash to include the current and parent directories:  $ ls -la total 2600 drwxrwxrwx   2 terdon terdon 2162688 Sep 10 16:22 . drwxr-xr-x 142 terdon terdon  491520 Sep 10 15:34 .. -rw-r--r--   1 terdon terdon       0 Sep 10 16:22 foo $ echo .* . ..   If I run rm -rf .* on my Debian using GNU bash, version 4.2.36(1)-release and rm from rm (GNU coreutils) 8.13, I get this message:  $ rm -rf .* rm: cannot remove directory: `.' rm: cannot remove directory: `..'   Is this a GNU thing or is it POSIX? Are there any *nix systems where the command above will silently delete . and ..?  Also, is this a safety feature of the shell or of the rm command itself? How to show the transfer progress and speed when copying files with cp? Otherwise, is there any alternative command line utility that can achieve this? How many solutions $X^{10} - I=0$ has in $M_2(\mathbb{R})$? How many solutions $X^{10} - I=0$ has in $M_2(\mathbb{R})$? Where $M_2(\mathbb{R})$ denotes the set of $2 \times 2$ real matrices.  I absolutely have no idea of where I should start from. $I$ and $-I$ solve the equation, but are there any other non-trivial solutions? Are there infinitely many solutions? How to automatically hyphenate within \texttt? This question was phrased as a bug report for LaTeX and I thought it would be helpful to be presented here (including an answer).  The person had a lot of "common words" in typewriter font sprinkled around his text and he was surprised that this text would not automatically hyphenate and he had to manually insert \- to mark hypenation points. As you know, we can enable the spell checker on single line text boxes in the following way:    We have to enable it every time for checking words.  How can we enable it forever, so it doesn't need to be enabled each time? Back transforming regression results when modeling log(y) I'm fitting a regression on the $\log(y)$.  Is it valid to back transform point estimates (and confidence/prediction intervals) by exponentiation?  I don't believe so, since $E[f(X)] \ne f(E[X])$ but wanted other's opinions.  My example below shows conflicts with back transforming (.239 vs .219).  set.seed(123)  a=-5 b=2  x=runif(100,0,1) y=exp(a*x+b+rnorm(100,0,.2)) # plot(x,y)  ### NLS Fit f &lt;- function(x,a,b) {exp(a*x+b)}  fit &lt;- nls(y ~ exp(a*x+b),  start = c(a=-10, b=15))  co=coef(fit) # curve(f(x=x, a=co[1], b=co[2]), add = TRUE,col=2,lwd=1.2)  predict(fit,newdata=data.frame(x=.7)) [1] 0.2393773  ### LM Fit # plot(x,log(y)) # abline(lm(log(y)~x),col=2) fit=lm(log(y)~x) temp=predict(fit,newdata=data.frame(x=.7),interval='prediction') exp(temp)         fit       lwr       upr 1 0.2199471 0.1492762 0.3240752 From a link from the question  to the HP Wikia, I started mulling over something I read about Voldemort and the "regeneration" potion (not a canon potion, but I'll use it for lack of a better way of describing it. . .) and would really like to know:  How did Voldemort drink the regeneration potion to turn his body into the grotesque childlike form we see in Goblet of Fire while he was still in a "mangled spectral state"?  Spectres (i.e. spirits or ghosts) are non-corporeal; they obviously can't consume food or drink liquids. So, how did Voldemort overcome this difficulty and manage to drink the regeneration potion? Home button broken and Notification Center stuck my home button on my ipod touch 5th Gen has been broken for  while so I have been using assistive touch. Earlier on my notification centre got stuck half way down the lock screen, making my ipod completely unusable.   I tried turning it off but the notification centre is blocking the slider and obviously I can't restart it since my home button is broken. Has anyone got any solutions? It's an absolute nightmare! A correct preposition - account closed due/by/at 12th of May How do I rename a directory via the command line? The set of convergent points of a sequence of continuous functions is Borel Suppose that $f_n$ are continuous.  Prove that $E=\{x\in \mathbb{R}:f_n(x)\text{ is convergent}\}$ is Borel.   My first instinct is to take balls around $\lim_{n\to\infty}f_n(x)$, for each $x$ in $E$, and then show that there has to be some ball small enough so that everything inside that ball is a member of $E$. But this would imply not only that $E$ is Borel, but moreover that it is open, so I don't think that can be right.  Can someone show me how to do this? How do you handle uneven tiles while rendering a tile map? Here is what I want to do with my tile map that I am unsure of. As you can see the top walls are way larger then the bottom and side ones (this is also an issue with my corners which are odd shapes [more like an L then a square] as well as larger then the 40x40px that my current tiles are).    I have also attached the tile sheet I am using for the tiles to show you what the corners and tops look like compared to the rest of the tiles.  What I’m thinking is I might be able to just draw them using the same array I have if I use the tiles width and height instead of a set WxH? I dont know how well this will work, but were the blocks that will have different values will be blocks the player/NPCs can’t step on then it may not be an issue, again I’m not even sure if that is a viable solution.  My code in case you want to see how I'm currently handling it:  (pastebin) (edit:im not handling it, meant how I'm currently doing the map)  tl;dr - how can I take take my 2d looking tile map and change it to have different size tiles? or is there a better way to do what I am trying to do? Is there a name of a method that I should be looking for while googling? I got a stomach flu or I got the stomach flu or I got stomach flu? When I googled stomach flu, there were many questions and statements like      How do you get the stomach flu?    Or     Difference between a stomach flu and food poisoning    Or      Six facts about stomach flu   When do you use the, a or nothing? And which should I use for a sentence like "I got a stomach flu from someone" ? "Fatal error: Class name must be a valid object or a string in C:\xampp\htdocs\uc\includes\common.inc on line 7506"  The error above showed up unexpectedly. I was adding a "Long text" field to a content-type, but when I expected to be brought to a saved field configuration page, I instead got a Fatal error on every page.  I had a look at includes\common.inc:  /**  * Get the entity controller class for an entity type.  */ function entity_get_controller($entity_type) {   $controllers = &amp;drupal_static(__FUNCTION__, array());   if (!isset($controllers[$entity_type])) {     $type_info = entity_get_info($entity_type);     $class = $type_info['controller class'];     $controllers[$entity_type] = new $class($entity_type); // Line 7506   }   return $controllers[$entity_type]; }   However, I just don't understand Drupal at a low enough level to understand the relationships between these variables and functions.  I also searched Drupal for people with a similar problem:  (1)  who claims that I have to install the submodule "Relation Endpoints Field", but searching Drupal, .  (2) :  DELETE FROM cache; DELETE FROM cache_filter; DELETE FROM cache_menu; DELETE FROM cache_page; DELETE FROM watchdog;   (via , misleading name, I know.)  Clearing those tables from the cache did not help.  (3) And another user . Because I can't find a module specifically pertaining to old search facets, I went into MySQL system table, and marked the Search module's line with a 0 status, indicating to stop loading the module altogether.  On that same link as (3), one user suggested enabling the , so I dropped all the tables, imported the latest stable .sql, and installed the module, but after recreating the problem (trying to add a field to a content-type), that as well concluded fruitless.  Still, the problem remains  The Fatal error shows up on every page, making the site completely non-functional. Any help or direction would be immensely appreciated. Thank you for looking.