Book recomendations for learning number theory.
Good Number Theory books to start with?
Is there a way to force a C# class to implement certain static functions? For PhD applications, I have mentioned that I am interested in some research areas. Besides, I am thinking about mentioning some professors' names, so that it shows I have read their websites and I am not just sending out generic statements applicable to many schools.  But is it good to mention some professors' names in the statement of purpose, if I don't know them personally? In what cases is it good, and in what cases it isn't?  For example, is it good to only mention    those whose research directions are aligned with my research interests, or/and those whose books or notes I have heard of, or better, might have read some parts of, or/and those who are already established and famous (such as having certain titles), or/and those that are directing the PhD graduate programs, or/and those that are heads of the departments?    What will other professors who are not mentioned think? How will you do, if it were you? Is the word "management" singular or plural? Presentation $\langle x,y,z\mid xyx^{-1}y^{-2},yzy^{-1}z^{-2},zxz^{-1}x^{-2}\rangle$ of group equal to trivial group Problem: Show that the group given by the presentation $$\langle x,y,z  \mid xyx^{-1}y^{-2}\, , \, yzy^{-1}z^{-2}\, , \, zxz^{-1}x^{-2} \rangle $$ is equivalent to the trivial group.   I have tried all sorts of manners to try to show that the relations given by the presentation above imply that $x=y=z=e$. However, I am stuck and would appreciate any hints as to how I should move forward. "It really doesn't matter" v "It doesn't really matter" How to enable ssh root access on Ubuntu 14.04 I´d like to enable the ssh root access on Ubuntu 14.04. Editing the /etc/ssh_config (PermitRootLogin yes) didn't affect anything. Is Multiplying the Inverse Better or Worse? When dealing with double data types is multiplying by the inverse better or worse? Which way is faster? Which way uses less memory? Which way is preferred? How does MSIL handle this? SquareInches = MMSquared / 645.16  SquareInches = MMSquared * 0.0015500031000062000124000248000496    NB: 10K users will note that this is a duplicate of , which was deleted because the original question asker decided to berate everyone in the 'comments' section of the question.  This question was reposted because it is a 'good' question. Please 'uncheck' Community Wiki for your answer, as I'm only posting this as CW so that it's not seen as a 'reputation' grab. Related Question: I know from reading  that the "primary" use of the IDisposable interface is to clean up unmanaged resources.  To me, "unmanaged" means things like database connections, sockets, window handles, etc.  But, I've seen code where the Dispose() method is implemented to free managed resources, which seems redundant to me, since the garbage collector should take care of that for you.  For example:  public class MyCollection : IDisposable {     private List&lt;String&gt; _theList = new List&lt;String&gt;();     private Dictionary&lt;String, Point&gt; _theDict = new Dictionary&lt;String, Point&gt;();      // Die, clear it up! (free unmanaged resources)     public void Dispose()     {         _theList.clear();         _theDict.clear();         _theList = null;         _theDict = null;     }   My question is, does this make the garbage collector free memory used by MyCollection any faster than it normally would?  edit:  So far people have posted some good examples of using IDisposable to clean up unmanaged resources such as database connections and bitmaps.  But suppose that _theList in the above code contained a million strings, and you wanted to free that memory now, rather than waiting for the garbage collector.  Would the above code accomplish that? The ring $ℤ/nℤ$ is a field if and only if $n$ is prime Let $n \in ℕ$. Show that the ring $ℤ/nℤ$ is a field if and only if $n$   is prime.   Let $n$ prime. I need to show that if $\bar{a} \neq 0$ then $∃\bar b: \bar{a} \cdot \bar{b} = \bar{1}$. Any hints for this ?  Suppose $ℤ/nℤ$ is a field. Therefore: for every $\bar{a} \neq 0$ $∃\bar b: \bar{a}\cdot \bar{b}=1$. How can I show that $n$ must be prime ? Are background jobs and foreground jobs always in running state? A job can be running or stopped (suspended).  When we are talking about foreground jobs and background jobs, are we only talking about running jobs?  Are background jobs and foreground jobs always in running state?  Does it make sense to say whether a stopped job is foreground or background? Let's say a 10N cube (in air, on Earth) rests flat on a scale at the bottom of a tank of water, and the scale reads 8N, so there is 2N of buoyant force on the cube.  How does the buoyant force manifest itself on the cube?  I assume the force is normal to the surface of the scale, but if there isn't any water under the cube, how does the water push up on the cube? How do I evaluate this integral $\int_0^\pi{\frac{{{x^2}}}{{\sqrt 5-2\cos x}}}\operatorname d\!x$? Show that $$\int\limits_0^\pi{\frac{{{x^2}}}{{\sqrt 5-2\cos x}}}\operatorname d\!x =\frac{{{\pi^3}}}{{15}}+2\pi \ln^2 \left({\frac{{1+\sqrt 5 }}{2}}\right).$$   I don't have any idea how to start, but . How do I remove/delete a folder that is not empty? Do photons lose energy due to gravitational redshift? If so, where does the lost energy go? In the gravitational redshift, the frequency of photons radiated from some source is reduced. As the energy of a photon is given by $\hbar\omega$, if the frequency is reduced where is the lost energy? I have a very simple question. When I do this:  \catcode`\@=0 \catcode`\\=11 @mycommand{a} @catcode`@\=0 \catcode`\@=11   It works perfectly and calls \mycommand correctly. However, when I try to define \mycommand with similar trickery:  \newcommand\mycommand[1]{   \catcode`\@=0   \catcode`\\=11   @textbf{Hello #1}   @catcode`@\=0   \catcode`\@=11 }   compilation stalls and gives me the * prompt. If I give it Ctrl+D, it terminates without producing any output. Indeed, it seems that changing the catcodes inside a command has no effect whatsoever, because the following works perfectly (whereas, from my understanding, it shouldn't):  \newcommand\mycommand[1]{   \catcode`\@=0   \catcode`\\=11   \textbf{Hello #1} }   Can anybody please explain to me what's happening? Combinatorics identity question I encountered this particular combinatorial identity :$$\begin{align}&amp;1\dbinom{n-1}{r-1}+2\dbinom{n-2}{r-1}+\cdots+(n+1-r)\dbinom{r-1}{r-1}\\=&amp;\dbinom{n}{r}+\dbinom{n-1}{r}+\cdots+\dbinom{r}{r}\\=&amp; \dbinom{n+1}{r+1}\end{align}$$How does one figure it out? Delay in approval leads to missing out on being awarded an "Explainer" badge I am not sure if this is a bug or a design error, but I came across this issue on Robotics. I assume that the same code is employed across the various SE sites, so I am posting my question here. I submitted an answer to  at 16:26.    I then suggested an edit to the question at 17:03. However, it was not until 20 hours later, at 13:03, that the edit was approved, so I appear to have missed out on the &quot;Explainer&quot; badge, as the edit should be done within 12 hours.    I don't know the reason for the delay, as I do not have a low approval rating:    I'm not bothered about missing out on the badge, nor am I criticising the reviewers - after all, we all do this in our free time. However, it does seem that the algorithm for the badge relies on the approval time, in lieu of the submit time, which does not seem to be correct, as the badge then becomes dependant upon the reviewer's availability/responsiveness, rather than upon the actions of the answerer/editor.  TL;DR Should the badge also check if the edit was caused by a suggested edit and, in those cases, look at the time it was suggested rather than the time it was eventually approved? What can cause "Discovered - currently not indexed" in the new GWT Why is this sentence correct?     She suggested that he go to the cinema.   I would definitely use goes instead of go. I was working on a Beamer presentation, and noticed that, if I open a group scope and use a \tiny, the font size will bet set, but there will be a large space between lines. However, if I issue a \par at the end of the paragraph, it works as intended:  \documentclass[t]{beamer} \newcommand{\shortlipsum} {Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Ut purus elit, vestibulum ut, placerat ac, adipiscing vitae, felis. Curabitur dictum gravida mauris. Nam arcu libero, nonummy eget, consectetuer id, vulputate a, magna. Donec vehicula augue euneque. Pellentesque habitant morbi tristique senectus et netus et malesuada fames ac turpis egestas. Mauris ut leo. Cras viverra}  \begin{document} \begin{frame} \shortlipsum  {\tiny \shortlipsum\par   % &lt;--- removing this \par will break line spacing } \end{frame} \end{document}   Here's a comparison between not using and using \par:    Am I doing it the right way? Why is \par the right way?