Number Theory: Prove that any primitive root $r$ of $p^n$ is also a primitive root of $p$
Prove that , any primitive root $r$ of $p^n$ is also a primitive root of $p$
Mavericks input source is set to US keyboard in login screen I'm from Belgium, and on my own user account, the input source is set to a Belgian (actually French) keyboard. The input sources menu is enabled in the menu bar, so I can switch it to the Belgian keyboard before entering my password, but how can I permanently change the input source used for login? I'm the system admin, and I know how to get root access if necessary.  It worked perfectly in Mountain Lion. How do I include lines in resolv.conf that won't get lost on reboot? I have finally migrated to 12.04 from 7.10. I have one last part to complete but I am stumped. I am using Puppet on each server, and in the past I have included a nameserver address and a search domain name for the puppetmaster in resolv.conf.   search puppetmaster.com nameserver 192.168.1.XXX   In 12.04 resolv.conf gets overwritten when rebooted. I cannot use a static IP for these, so using the /etc/network/interfaces to help me out is a nill point.  # Dynamic resolv.conf(5) file for glibc resolver(3) generated by resolvconf(8) #     DO NOT EDIT THIS FILE BY HAND -- YOUR CHANGES WILL BE OVERWRITTEN nameserver 127.0.0.1   Is there a way to get resolvconf to handle this either in the head, tail or base? If there is, are there any examples I can use to tweak on my server.   Any help is much appreciated. What could cause exposure problems after my camera got wet? I had an unfortunate accident where my DSLR camera (and me) briefly ended up in a river. The camera wasn't fully submerged but it did get very wet. It still works, sort of, although the electrics are doing some random things. I'm hoping that's just a bit of dampness doing that, that will fix itself when it dries out fully.  More concerning is that the documents exposure metering seems to be all wrong when using the viewfinder. Pictures are extremely overexposed and nearly white. But if I use live view, then they are fine.  Why could cause this? I assume the light is routed a different way when using the viewfinder and maybe it is going through something with water/condensation on it? What ultimates can be damage boosted by Mercy? What ultimates can be damage boosted by Mercy?  We already know that , but are there other ultimates that can be damage boosted by Mercy? Can all ultimates be damage boosted by Mercy? How can I put two figures side-by-side? Not two sub-figures, but two actual figures with separate "Fig.: bla bla" captions.  A figure is supposed to spread over the entire text width, but I have two figures which are narrow and long, and I need to save the space in order to withstand the pages limit. What is an efficient way to implement a singleton pattern in Java? how to convert NTFS to ext4 in windows? Is the C# "explicit implementation" of the interface present in Java? In C#, if you have two base interfaces with the same method (say, F()) you can use explicit implementation to perform different impl. for F(). This alloes you to differently treat the object, corresponding to the current point of view: as IMyInterface1 or IMyInterface2. Is this possible in Java? Why are my render times 30+ minutes, but other people's are ~3 minutes? Computing the $n^{\textrm{th}}$ permutation of bits. I've seen  post about the $n^{\textrm{th}}$ permutation of a set but that is not what I need. If you have a bit string (ones and zeros only) there are  to quickly permute the NEXT lexicographically ordered bit permutation. For example take $$000111 \rightarrow 001011 \rightarrow 001101\rightarrow\cdots$$ etc. If the string is long then there are going to be approximately one Bajillion of these things. And I want to know how to compute the $n^{\textrm{th}}$ guy without doing the exhaust.  Backgound: This is for a parallel computing job where I need to farm out the search space of elements of a set.  Foreground: I own a copy of Knuth's 4th volume of "The Art of Computer Programming" and I think the answer is in there but I can't seem to find it. (It's like 900 pages).  I'm posting here in the hopes that someone has knowledge of this (obviously). Even if you can point me to a source, say the part in Knuth's book where he describes this problem I would be most grateful. From injective map to continuous map Fantasy novel with amnesiac witch, impotent soldier and cursed animals The story is set in a vaguely medieval world in which a young woman has been cursed by the evil witch who has enslaved her and caused her to lose her memory. With the aid of her talking animal friends who have also been cursed (toad has a huge stone under his tongue meaning he can't feed and a crow with a gem embedded in its wing and there's a little grey kitten too who never grows up) she sets out on a journey to break the curse for herself and her friends. Along the way she meets an injured unicorn and a horny-but-moral soldier rendered impotent by a curse after refusing the disguised witch's sexual advances.  After adventures there's a 'happy ever after' for all of the heroes and the witch meets a sticky end.    In the UK the book had a deep red cover with an illustration. I read it in either the 1990's or the early 2000's. Cracking book, definitely for adults given some of the situations, language and humour. Why “which” returns none on some commands I can run? Why should I use the keyword "final" on a method parameter in Java? I can't understand where the final keyword is really handy when it is used on method parameters.    If we exclude the usage of anonymous classes, readability and intent declaration then it seems almost worthless to me.   Enforcing that some data remains constant is not as strong as it seems.    If the parameter is a primitive then it will have no effect since the parameter is passed to the method as a value and changing it will have no effect outside the scope.  If we are passing a parameter by reference, then the reference itself is a local variable and if the reference is changed from within the method, that would not have any effect from outside of the method scope.    Consider the simple test example below. This test passes although the method changed the value of the reference given to it, it has no effect.  public void testNullify() {     Collection&lt;Integer&gt; c  = new ArrayList&lt;Integer&gt;();           nullify(c);     assertNotNull(c);            final Collection&lt;Integer&gt; c1 = c;     assertTrue(c1.equals(c));     change(c);     assertTrue(c1.equals(c)); }  private void change(Collection&lt;Integer&gt; c) {     c = new ArrayList&lt;Integer&gt;(); }  public void nullify(Collection&lt;?&gt; t) {     t = null; } I need to print an image at a certain size. What dimensions and resolution should I use? How to select (highlight) text in Skype chat without using a mouse? Recently I started learning to use keyboard shortcuts almost for everything I'm doing on the laptop, but soon enough after I started, I bumped into the problem of being unable to highlight (select) simple text in Skype chats.  I did some research and I couldn't find any solution to my problem. Have I missed an already answered question that is the same like mine or if not what solution can you suggest in order to achieve the results I want without touching mouse or touchpad or any cursor moving device? What part of speech is “there” when used in “There is (blah blah)”? What part of speech is there in the sentence “There is a book on the table?”  Also, while typing it out, another question pops up vis-à-vis punctuation. In my complete first sentence above, I ended it with a question mark since my main sentence is a question. The quote is not a question, but it looks like it because of the question mark. Is this the right way to punctuate? Composition of two functions - Bijection I keep going around in circles with this problem:  Given the functions $f:X\to Y$ and $g:Y\to Z$, if $g\circ f:X\to Z$ is bijective, then prove $f$ must be one-to-one and $g$ must be onto. I was able to prove this without difficulty. However the second part of the problem has me stumped:  a) Find functions $f,g$ where $g\circ f$ is bijective but $f$ fails to be onto.  b) Find functions $f,g$ where $g\circ f$ is bijective but $g$ fails to be one-to-one.  My attempt for (a): Let $f:\mathbb{R}\to \mathbb{R}^+$ be defined by $f(x)=e^x$ and $g:\mathbb{R}^+\to \mathbb{R}$ be defined by $g(x)=\ln(x)$. Then $g\circ f(x)=g(f(x))=g(e^x)=\ln(e^x)=x$ which is bijective, $g$ is onto and $f$ fails to be onto... or so I thought. Upon closer inspection $f$ $\underline{is}$ onto for the given codomain $\mathbb{R}^+$.  And now I'm stuck in a rut. Anyone out there have any hints? How to simulate slow internet connections on the mac What's a good software solution to slow down internet connections on the mac for testing and simulations?  My primary need is when testing code in the iPhone / iOS simulator, so slowing down the connection for specific applications or processes would be great. Why does adding a DOCTYPE affect my CSS?