Let $p$ be prime with $p\equiv 2 \bmod 3$. Show that for any integer $a,p\nmid a$ there is an integer $x:x^3 \equiv a\bmod p$
Let $p$ be a prime number, and let $\mathbb{F}_p$ be the field with $p$ elements. How many elements of $\mathbb{F}_p$ have cube roots in $\mathbb{F}_p$?   I had this question on an exam and after reviewing I am still not sure. Any help would be appreciated.
Enum ToString with user friendly strings My enum consists of the following values:  private enum PublishStatusses{     NotCompleted,     Completed,     Error };   I want to be able to output these values in a user friendly way though. I don't need to be able to go from string to value again. With Glade Designer that lets you set up Gtk-Ui's you can check the option that windows are not shown in the launcher. If I take a look at the ui-file generated by Glade, it says &lt;property name="skip_taskbar_hint"&gt;True&lt;/property&gt;. In gtk code this hint is  window.set_property('skip-taskbar-hint', True). However I believe that this option is not exclusively Gtk-specific, and want to apply this one on my PyQt4-app. The Ubuntu window manager must be able to receive that hint, otherwise I would see the window of a gtk-application i use (stickynotes-indicator). So there MUST be a way of doing this in Qt or any other language, right?  Does anybody know such an option? Python or Qt syntax, doesn't matter! Direction of $d\mathbf{l}$ I'm new to this website. I've checked the other topics and couldn't find the answer to my question. I've also tried to find the result with TI-CAS but the only thing it does is simplify the inequality which really doesn't help at all.  the equation is : $1000n^3 &lt; 2^n$ Now I've been trying to use log but it gets me nowhere (I haven't seen this form before), the answer is supposed to be 24 but I'm not sure. So my question is how to solve step by step this inequality for n. Could you guys help me with this one? If A is an infinite set and B is at most countable set, prove that A and $A \cup B$ have the same cardinality I am trying to prove: If A is an infinite set and B is at most countable set, prove that A and $A \cup B$ have the same cardinality.  Set definitions here are from baby Rudin.   Attempt: There are 4 possible cases:  1.A is countable and B is finite.  2.A is countable and B is countable.  3.A is uncountable and B is finite.  4.A is uncountable and B is countable.  Then for each case I need to show that $A \cup B \sim A$. I can prove 1-3, but I am unsure how to prove case 4. And in general, it seems that my proof strategy is inefficient and there exist much faster proof then proof by cases. How to learn commutative algebra? Comparing currency values as floats does not work Opening PDF generated by Python AddIn using Report (*.rlf) file automatcally? As a test I have:   Authored a test.mxd with one layer called Localities in a single data frame called Layers Authored a test.rlf using View | Reports > Create Report of ArcGIS 10.2.2 for Desktop Created a Python AddIn which creates the expected PDF file from the Report (*.rlf).   This is the code which all works to create the PDF report from the Localities features within the current extent:  import arcpy  import pythonaddins  class ReportForExtent(object):     def __init__(self):         self.enabled = True         self.checked = False      def onClick(self):         mxd = arcpy.mapping.MapDocument("CURRENT")         df = arcpy.mapping.ListDataFrames(mxd,"Layers")[0]         lyr = arcpy.mapping.ListLayers(mxd,"Localities",df)[0]         arcpy.mapping.ExportReport(lyr,r"C:\temp\test.rlf",r"C:\temp\test.pdf","EXTENT",extent=df.extent)         del mxd   Is there a way to have the Python AddIn not just create the PDF file but also open it in Adobe Acrobat Reader? This is something that always annoys me when putting an A4 letter in a oblong envelope: one has to estimate where to put the creases when folding the letter. I normally start from the bottom and on eye estimate where to fold. Then I turn the letter over and fold bottom to top. Most of the time ending up with three different areas. There must be a way to do this exactly, without using any tools (ruler, etc.). Why getting this error tlmgr Unknown directive? I haven't used tlmgr for a while but when I tried to install any package with it today, It returned following message:     (running on Debian, switching to user mode!) Unknown directive ...containerchecksum c59200574a316416a23695c258edf3a32531fbda43ccdc09360ee105c3f07f9fb77df17c4ba4c2ea4f3a5ea6667e064b51e3d8c2fe6c984ba3e71b4e32716955... , please fix it! at /usr/share/texlive/tlpkg/TeXLive/TLPOBJ.pm line 210,  line 5579.   I've tried to use fix specified on :     You have to download update-tlmgr-latest.sh from the tlpretest repository   and run it. Afterwards you will be able to work with tlmgr as usual   (announced by Norbert a few hours ago).   I've downloaded it from , but it returned another error:  Verifying archive integrity... All good. Uncompressing TeX Live Manager Updater............................................................................................................................................................................ ./runme.sh: Cannot find TeX Live root using kpsewhich --var-value=SELFAUTOPARENT. ./runme.sh: Please set your PATH as needed, otherwise it's hopeless.   Minimum example  Install cleanly vanilla TeXLive 2016 with tmlgr after having old TeXLive in the system successfully: there is no errors/warnings in /usr/local/texlive/2016/*.log, complete log example . Run the following but you get the same output as above when trying to update the system  root@masi:/home/masi/Downloads/t/install-tl-20161019# tlmgr update --self (running on Debian, switching to user mode!) Unknown directive ...containerchecksum c59200574a316416a23695c258edf3a32531fbda43ccdc09360ee105c3f07f9fb77df17c4ba4c2ea4f3a5ea6667e064b51e3d8c2fe6c984ba3e71b4e32716955... , please fix it! at /usr/share/texlive/tlpkg/TeXLive/TLPOBJ.pm line 210, &lt;$retfh&gt; line 5579. Does an adjective or longer modifier apply to both nouns when they are joined with 'and' [or 'or']? reusing pthread_t variable for currently running threads MySQL: Allow both remote AND local connections Prove that the number of $\alpha\in\mathbb{F}_{27}$ such that $|A_\alpha|=26 $ equals 12. Let $\mathbb{F}_{27}$ denote the finite field of size 27. For each $\alpha\in\mathbb{F}_{27}$ , define $ A_{\alpha}$ $=$ { $ 1, 1+\alpha , 1+\alpha+{\alpha}^2 , 1+\alpha+{\alpha}^2+{\alpha}^3 , .... $} . Then  prove that  the number of  $\alpha\in\mathbb{F}_{27}$ such that $|A_\alpha|$=26 equals 12.  $ 0\in A_\alpha $ if and only if $\alpha\neq 0$ For the first proof , the multiplicative group of the field $\mathbb{F}_{27}$ is cyclic and its order is 26  , so there are 12 generators in that group.   Now  I need  to  show  that  $ 1+\alpha+{\alpha}^2+.. {\alpha}^{25} $ =0  , where order of $\alpha$=26 . Since   $\alpha ^{26}$ =1 ($\alpha$-1)(1+$\alpha$+${\alpha}^2$+...+${\alpha}^{25}$)=0 , since $\alpha\neq 1$  , So $ 1+\alpha+...+{\alpha}^{25} $ =0 My question is , can I write these last two  lines?  I mean are those two lines alright to write? If those two lines are invalid then I need some help to show  $ 1+\alpha+{\alpha}^2+..+{\alpha}^{25} $=0  .Thanks Find the solution of following differential equation:     $$\frac{dy}{dx}=\frac{1}{xy(x^2 \sin (y^2)+1)}$$   Could someone hint me something to get through this problem? Career Change in to Testing I've around 10+ years of experience in Banking operations including  reconciliations, sales, customer services, asset management, and financial transaction QA.  Coming to academics I've done my PG in commerce (costing, auditing, banking, and statistics) also a diploma in project management, ISTQB foundation, Prince 2 foundation and ITIL foundation.    Can any one advise me if the testing role is ideal for me and how to start with it. 32 bit or 64 bit OS on a 512 mb VPS? I am planning to rent a VPS service with following basic specification:   512mb of RAM 1 CPU 20gb HDD (4gb of it is swap) Ubuntu 10.10 server   I am able to choose between a 32 bit or 64 bit install of Ubuntu server as the operating system.   My question is would it be better to chose a 32 bit operating system since I will have only 512mb of RAM, rather than using a 64 bit operating system and dealing with the additional overheads?  edit:  I have installed Ubuntu server 10.10 32 bit and 64 bit on VirtualBox. After a minimalistic installation with only ssh server, the 32 bit installation consumes 63mb of RAM when first boot up and 64 bit consumes 85mb of RAM.  After installing and running  daemon, 32 bit consumes 99mb of RAM when first boot up and 64 bit consumes 128mb. A subring of the field of fractions of a PID is a PID as well. Let $A$ be a PID and $R$ a ring such that $A\subset R \subset \operatorname{Frac}(A)$, where $\operatorname{Frac}(A)$ denotes the field of fractions of $A$. How to show $R$ is also a PID?   Any hints? The process substitution &lt;(cat) doesn't work. Why?  head &lt;(cat)  cat: -: Input/output error   I can use an extra file descriptor, but I don't understand why the above doesn't work.  # Using file descriptor 3 to reroute the input (head &lt;(cat &lt;&amp;3)) 3&lt;&amp;0 Let $L,M$ two subspaces of the vector space, $V$ such that both $L,M \ne V$.   Prove: $L\cup M \ne V$.     I think this is a case of a proof by contradiction. Lets assume $L \cup M = V$.   Hence, $$\dim(V) = \dim(L) + \dim(M) - \dim(L\cap M)$$  How to proceed?