How do I solve this equation: $$x^5 \equiv  3\pmod{19} $$ where all I know is Fermat's Little Theorem and operations on multiplicative group $\mathbb Z_p^*$?
We know that $221 = 17*13$. So we can check if the system has roots to both of those equations separately, which it does:  $x^{5} \equiv 2$ mod $13$ has the solution $6 + 13n$ and $x^{5} \equiv 2$ mod $17$ has the solution $15 + 17n$.   I got these numbers from wolfram, I have no idea how to solve this problem WITHOUT a calculator. And even after finding these numbers. How would one obtain a solution modulo $221$? I was thinking the Chinese Remainder Theorem but I am under the assumption that CRT only applies to problems with powers of $x$ which are $1$.  Thanks.
Sometimes, when I upgrade a system via do-release-upgrade, the update process fails and the script tells me to finish the process manually via dpkg --configure -a. This is a not serious problem - after manually solving the dependencies problem and the finishing upgrade process all works well.  But the do-release-upgrade script has a cleanup process after finishing the upgrade:  Remove obsolete packages?   XXX packages are going to be removed.   Continue [yN]  Details [d]y   which removes some obsolete packages from old version of system.  Seems that this process doesn't execute when the automatic upgrade fails and I finish upgrade manually, so obsolete packages from old version remain installed in the system.  How I can run the "Remove obsolete packages" process manually after finishing the upgrade by hand? Why does the volume of the unit sphere go to zero? The volume of a $d$ dimensional hypersphere of radius $r$ is given by:  $$V(r,d)=\frac{(\pi r^2)^{d/2}}{\Gamma\left(\frac{d}{2}+1\right)}$$  What intrigues me about this, is that $V\to 0$ as $d\to\infty$ for any fixed $r$.  How can this be?  For fixed $r$, I would have thought adding a dimension would make the volume bigger, but apparently it does not.  Anyone got a good explanation? How to address a professor in letter? I am writing a document using LaTeX and generating PDF.  Can anybody tell me how I can generate the PDF metadata, e.g. author or title, from LaTeX?  On Windows XP, I am using MikTex's texify to generate the PDF.  On linux, I am using pdflatex to generate the PDF.  Any answers for either platform would be much appreciated! How to add the name of the supervisor in a @thesis field? In the biblatex doc, it says ( Version 1.7, p.11):     thesis      A thesis written for an educational institution to satisfy the   requirements for a degree. Use the type field to specify the type of   thesis.      Required fields: author, title, type, institution, year/date      Optional fields: subtitle, titleaddon, language, note, location,   month, isbn, chapter, pages, pagetotal, addendum, pubstate, doi,   eprint, eprintclass, eprinttype, url, urldate   I would like to know if there is any "canonical" way to add the name of the supervisor(s) in a field of the .bib file (no .bbl manipulation please, I want my bibliography to be usable "as it" in several documents).  The expect result could be something like  :     Meand Myself. “A brilliant result, explained”. PhD thesis**, under the   joint supervision of Pr. Toto and Pr. Tata**. Université Paris 24,   Nov. 2013.   The "under the (joint) supervision" could be a string that would be declined in four ways:   under the supervision of  under the joint supervision of and the abridged forms Easy is to beginner as medium is to I want to display three levels in my game...but I don't know what the text should say.  easy -&gt; Beginner  medium -&gt; ?  hard -&gt; Expert   Basically, I want the form that describes the person playing the game, instead of the game itself. What adjective is best?  I have considered adept.  I doubt my users will know what adept means. Also, I need something in between expert and beginner. Adept falls into the expert category. Is there anything more common that I can use, that has the same impact as the other two words I mentioned?  Proficient would work, theoretically, but in day-to-day term, the levels "beginner - proficient - expert" won't sound great. I'm using an Android smartphone that I purchased Minecraft PE on; now I want to transfer my world to a iPhone. I have also purchased Minecraft PE on my iPhone. unable to change php version on cli How to use java.util.Scanner to correctly read user input from System.in and act on it? How to set a command block with a command inside using the setblock command Show some indication that account is banned from asking new questions How can I update my OpenGL to 3.0 or higher Equation of a rectangle I need to graph a rectangle on the Cartesian coordinate system. Is there an equation for a rectangle? I can't find it anywhere. A sum of series problem: $\frac{3}{1!+2!+3!} + \frac{4}{2!+3!+4!} + \cdots + \frac{2008}{2006!+2007!+2008!}$ I have a question regarding the sum of this series: $$\frac{3}{1!+2!+3!} + \frac{4}{2!+3!+4!} + \cdots + \frac{2008}{2006!+2007!+2008!}$$ My approach: I found that this sum is equal to: $$\sum_{n=3}^{2008}\frac{n}{(n-2)!+(n-1)!+(n)!}$$  I reduced it to : $$\sum_{n=3}^{2008}\frac{1}{n(n-2)!}$$  Please suggest how to proceed further. How to add “jQuery” onclick functionality for a column each row in jQgrid to display different popups for different rows I just can't seem to figure this one out.  I want to use the command line to connect to a MySQL database residing on another server.  I went ahead and created the username and password for the user.  I have also granted all privileges on that user for that database.  When using the command: mysql -h &lt;hostname&gt; -u &lt;username&gt; -p, I get the following error:  ERROR 1045 (28000): Access denied for user '&lt;username&gt;'@'&lt;local_machine_hostname&gt;' (using password: YES)   The problem is that it keeps appending the current machine's hostname into the username.  Obviously, that user@&lt;local_machine_hostname&gt; is not correct.  It doesn't matter what I type.  For instance, if I type:  mysql -h &lt;hostname&gt; -u '&lt;username&gt;'@'&lt;hostname&gt;' -p   It does the same, only in the error output, it says:  Access denied for user '&lt;username&gt;@&lt;hostname&gt;'@'&lt;local_machine_hostname&gt;'   Is there a setting in a configuration file which is allowing this to happen?  It's really quite annoying.  I need to set up a tikiwiki server, and it cannot connect because during the step where you set up mysql, it keeps appending the local machine's hostname to the mysql login name. find exclude directory How to read contacts on Android 2.0 Is there an Extended Backus–Naur form (EBNF) that covers all of English? What is an quick way to create a new blank file in Lion? In Windows, one can create a new file using Right Click -> Add New -> Type of File. Is there an analogous action in Lion?  I just want to create a new blank file on my Desktop.