Let p be a prime and let $a \in \mathbb Z$. Prove that the congruence $x^x \equiv a\pmod p$ is solvable.
What theorems can I use to prove this?
Is there a size of rectangle that retains its ratio when it's folded in half? A hypothetical (and maybe practical) question has been nagging at me.  If you had a piece of paper with dimensions 4 and 3 (4:3), folding it in half along the long side (once) would result in 2 inches and 3 inches (2:3), which wouldn't retain its ratio. For example, here is a piece of paper that doesn't retain its ratio when folded:   Is retaining the ratio technically possible? If so, what is the side length and ratio that fulfills this requirement? Any help would be appreciated.  Update:  I added "once" because I got an answer saying that any recectangle would work, as any rectangle folded twice has the original ratio. Nice answer, but not quite what I was looking for. As for the other answers, I got 3x as much information as I needed! Thanks! Strange behaviour of "ls -a | grep ^\." I wanted to list the content of a pwd and display only file starting with dot. I tried ls -a | grep ^\. but I cannot figure out why the output contains also the files which do not start with dot. For example:  Pictures .pip .pki .profile projects Public   I know that I can achieve what I want with ls -ld .* I am just curious about this behaviour of grep which I can't explain. I'm trying to make a logo that will be on the header of a website. What I'd like to achieve, and the reason I'm using Blender, is the projected shadows on the background looking perfectly real (as opposed to doing a drop-shadow in Photoshop).  The only thing is, the background has to be transparent because the website might use different background colors / gradients that make it impossible to match with the exported 3D background.  I made this dummy file (not the real logo... duh!):    If I use the file as is, the grey borders of the image will be impossible to match with the website's background color.  Here what I want:    So obviously, I tried masking the plane, which made the shadows disappear. I tried having a plane set to glass, which didn't work either.  And I looked around for a similar question, but didn't find one.   I am not trying to render , or a . I've seen  which has a very difficult solution to this problem, and  that is very similar but in Blender Internal.  is similar too but has never been solved. I'm only a beginner at Compositor, and I'm trying to learn more any chance I get, so any solution in Compositor is welcome if detailed enough. Although if there's a simpler or faster way, it's welcome too!  .blend file: Alright so I've seen some postings about loop cuts and done some research. I am very new to blender and modeling overall. I am trying to loop cut this concrete barrier. Now I'm following a tutorial and kind of did my own thing in some areas. I cleaned it up following some instructions in another answer here on stack exchange.  I can't for the life of me figure out why I can loop cut around the whole bottom:   ...but not the whole middle:   I see this marked as answered by another question. That question refers to ngons and inside faces. Now once again I'm totally new to this. I looked on the inside. I don't believe I have inside faces or any ngons by the terms I've read online in my research.  Here is a wireframe picture and the inside view:     Maybe I'm missing something the answers on other posts are saying. If so and you know what it is I'd be very appreciative of what it is. Exclude lstinputlisting I want to save paper when I print my LaTeX file. That's why I want to hide all my code, inserted wtih lstinputlisting{}  I already know how to exclude tables and images or math but the lstinputlisting{} is a problem.   Hope someone can help me  \usepackage{comment} \excludecomment{figure} \let\endfigure\relax \excludecomment{table} \let\endtable\relax \excludecomment{align*} \let\endalign\relax \excludecomment{lstlisting} \let\endlstlisting\relax   edit: added lstlisting Direction of friction for object that rolls with slipping Connected set in $\mathbb{S}^1$ Let $A$ be a connected set in $\mathbb{S}^1$. Show that $A^c$ is also connected.  My idea: suppose $A^c$ is not connected. Then, there exist $U,V \subset \mathbb{S}^1$ open and disjoint such that $U \cup V =A^c  $. Therefore $\mathbb{S}^1 = A \cup U \cup V$. If I had that $A\cup U$ was open I could arrive at  contradiction, since $\mathbb{S}^1$ is connected, but I can't say that... Precedence of user and group owner in file permissions Maintain the aspect ratio of a div with CSS I am doing some challenges. This is one them. I am trying to brute-force 4 digit pin with the password to get my desired answer. After connecting to the port It prompts me to enter the password then space then 4 digit pin. I tried to brute-force the pin using the script:  #!/bin/bash   nc localhost 30002   sleep 2    for i in {0000..9999};       if [[ $(echo 'UoMYTrfrBFHyQXmg6gzctqAwOmw1IohZ $i' &lt;/dev/stdin) = ^Wrong*]];                         then          continue       echo '[+] Pincode Cracked! Pincode = $i'       fi   done  but it seems that this doesn't input the pass and pin to stdin, before i tried doing something like this -&gt; if [[ $(echo 'UoMYTrfrBFHyQXmg6gzctqAwOmw1IohZ $i') = ^Wrong* ]]; What am I doing wrong here? UPDATE: Okay, so after researching around. I wrote this: for i in {0000..9999} do     if [ (echo &quot;UoMYTrfrBFHyQXmg6gzctqAwOmw1IohZ $i&quot; | nc localhost 30002 | grep -o Wrong) == &quot;Wrong&quot; ]    then       sleep 0.1       continue    fi    echo &quot;- - - - - - - - - - - - - - - - - - - - - - - - [$i]&quot; done  This might even work but as you can see it opens new connections in the loop which makes it really slow and exhaust the system. How to count occurrences of one field grouped by values of another? Is there a way of calculating the number of unique text strings in a field in ArcGIS 10.1? I have a number of repeated strings in this field, but would like to calculate the number of unique strings in a field based on groupings from another field.  E.g  - Current table:  Field 1    Field 2    Apple      Green      Apple      Red        Apple      Red Citrus     Yellow Citrus     Green Citrus     Orange Citrus     Orange   Result:  Field 1    Field 2 Apple      2 Citrus     3 Updated title of arxiv not recognised by Google Scholar? I submitted a paper to a journal and one of the referees suggested to slightly change the title of the paper. The paper was already posted on arxiv. I have updated the paper on arxiv after the revision, but this update is not recognised by Google Scholar. The main problem with this is that Google Scholar is not detecting the new citations to the arxiv paper since the new citations use the new title.  Any suggestions on how to solve this issue?  Update: I updated the arxiv version 2 months ago but Google Scholar does not detect it. It detects new papers that cite my manuscript, but not the updated arxiv.  Disclaimer. I am aware of the following questions, but they are related to different issues. Let $\alpha,\beta,\gamma$ be the roots of $x^3-3x^2+1 = 0$. Prove that $$\alpha^n+\beta^n+\gamma^n \equiv 4^n+5^n+(-6)^n \pmod{17}$$ where $n$ is an integer.   It sort of makes sense why they are congruent since we can say $x^3-3x^2+1 \equiv (x-4)(x-5)(x+6) \pmod{17}$, but since $\alpha,\beta,\gamma$ aren't integers, how do we prove this? First off I'll let you know I'm a total noob at blender I've used 3ds Max up until this point, but the company I'm working for at the moment needs me to use blender so I'm trying to make the transition over.  Literally all I'm making at the moment is a simple book and cannot understand how the bevel tool isn't working similar to how the chamfer tool would in 3ds Max.  I want a nice curved edge for the spine of the book however when I press control + B and try to bevel it bevels the edge unevenly with the new bevel pushing forward.    Is there a way I can get it to bevel evenly like it does in Max? I came across a list of functions in my calculus textbook whose indefinite integral cannot be found. It was written that the integral $$\int \frac{\sin(x)}{x} dx$$ cannot be evaluated without any explanation as to why.  I did some research over the internet and found out that the definite integral $$\int_{0}^{\infty} \frac{\sin(x)}{x} dx$$ can be evaluated using Laplace transformation and is equal to $\pi /2$. But I still couldn't find answer to my original question. I read somewhere that the integral cannot be expressed using 'elementary functions'. A little help is appreciated, I'm in Calc 1 going advanced than my course but I am sorry if my post shows lack of research. Thank you! Connecting to sql server database mdf file without installing sql server on client machine? Difference between "Should" vs "ought to" It's essential that the documents (should)/(ought to) be destroyed immediately.   Which one is more suitable one as I know "ought to" and "should" are synonyms. Subtracting 2 lists in Python Right now I have vector3 values represented as lists. is there a way to subtract 2 of these like vector3 values, like  [2,2,2] - [1,1,1] = [1,1,1]   Should I use tuples?  If none of them defines these operands on these types, can I define it instead?  If not, should I create a new vector3 class? One of my columns is called from. I can't change the name because I didn't make it. Am I allowed to do something like SELECT from FROM TableName or is there a special syntax to avoid the SQL Server being confused? A guide to moderating chat Chat is an offshoot of the main site, and using it is  during their time here.  Chat is our place for:        Real-time collaboration   Meeting fellow members of the community   Less structured, casual (but still roughly on-topic) conversation      Our chat system exists to serve the main site; users of each site can gather and commune with each other.  When in chat, we expect all users to abide by our  policy, which means that we expect community members to treat each other with respect.    While it is a place where the community can go to interact on a more casual level, chat still needs to be moderated by the community and there are tools in place to help with the moderation. This raises a few questions, that are addressed here: