Show that it is impossible that $P(a)=b$, $P(b)=c$, and $P(c)=a$ at the same time. Let $a, b, c$ be distinct integers, and let $P$ be a polynomial with integer coefficients. Show that it is impossible that $P(a)=b$, $P(b)=c$, and $P(c)=a$ at the same time.
Polynomial $P(a)=b,P(b)=c,P(c)=a$ Let $a,b,c$ be $3$ distinct integers, and let $P$ be a polynomial with integer coefficients.Show that in this case the conditions $$P(a)=b,P(b)=c,P(c)=a$$ cannot be satisfied simultaneously.  Any hint would be appreciated.
How do I count the number of occurrences of a char in a String? I have the string   a.b.c.d   I want to count the occurrences of '.' in an idiomatic way, preferably a one-liner.  (Previously I had expressed this constraint as "without a loop", in case you're wondering why everyone's trying to answer without using a loop). What are the use cases of SO_REUSEADDR? I have used SO_REUSEADDR to have my server which got terminated to restart with out complaining that the socket is already is in use. I was wondering if there are other uses of SO_REUSEADDR? Have anyone used the socket option for other than the said purpose? How can I pad an integer with zeros on the left? What's the Linux equivalent to Windows batch files? I have a batch file which I use to create an .apk file on my Windows machine. Now I need to be able to create the .apk file in Ubuntu but I don't know how translate my .bat file to a script to be able to run it on Ubuntu.  Below is the batch file which works fine on Windows.  Will you please give me some hints on how I can run it on Ubuntu?  @echo off set PAUSE_ERRORS=0 :user_configuration  :: Path to Flex SDK  set FLEX_SDK=C:\sdk\flex_sdk_4.5.1.21328  :: Path to Android SDK set ANDROID_SDK=C:\sdk\android  :validation if not exist "%FLEX_SDK%\bin" goto flexsdk if not exist "%ANDROID_SDK%\platform-tools" goto androidsdk goto succeed  :validation if not exist "%FLEX_SDK%\bin" goto flexsdk if not exist "%ANDROID_SDK%\platform-tools" goto androidsdk goto succeed  :flexsdk echo. echo ERROR: incorrect path to Flex SDK echo. if %PAUSE_ERRORS%==1 pause exit  :androidsdk echo. echo ERROR: incorrect path to Android SDK in 'bat\SetupSDK.bat' echo.  if %PAUSE_ERRORS%==1 pause exit  :succeed set PATH=%PATH%;%FLEX_SDK%\bin set PATH=%PATH%;%ANDROID_SDK%\platform-tools  :: Android packaging set AND_CERT_NAME="PeymanApp" set AND_CERT_PASS=fd set AND_CERT_FILE=cert\SampleApp.p12 set AND_ICONS=icons/android  set AND_SIGNING_OPTIONS=-storetype pkcs12 -keystore "%AND_CERT_FILE%" -storepass     %AND_CERT_PASS%  :: Application descriptor set APP_XML=application.xml  :: Files to package set APP_DIR=bin set FILE_OR_DIR=-C %APP_DIR% .  :: Your application ID (must match &lt;id&gt; of Application descriptor) set APP_ID=air.com.doitflash.SampleApp  :: Output packages set DIST_PATH=dist set DIST_NAME=PeymanApp  :validation %SystemRoot%\System32\find /C "&lt;id&gt;%APP_ID%&lt;/id&gt;" "%APP_XML%" &gt; NUL if errorlevel 1 goto badid goto end_validation  :badid echo. echo ERROR: Application ID (APP_ID) does NOT match Application descriptor '%APP_XML%'     (id)  echo.  :end_validation  set TARGET= set PLATFORM=android  ::call bat\Packager.bat if "%PLATFORM%"=="android" goto android-config  :android-config set CERT_FILE=%AND_CERT_FILE% set SIGNING_OPTIONS=%AND_SIGNING_OPTIONS% set ICONS=%AND_ICONS% set DIST_EXT=apk set TYPE=apk goto start  :start if not exist "%CERT_FILE%" goto certificate :: Output file set FILE_OR_DIR=%FILE_OR_DIR% -C "%ICONS%" . if not exist "%DIST_PATH%" md "%DIST_PATH%" set OUTPUT=%DIST_PATH%\%DIST_NAME%%TARGET%.%DIST_EXT% :: Package  echo true echo. call adt -package -target %TYPE%%TARGET% %OPTIONS% %SIGNING_OPTIONS% "%OUTPUT%"     "%APP_XML%" %FILE_OR_DIR% -extdir lib/ echo. if errorlevel 1 goto failed goto end  :certificate echo Certificate not found: %CERT_FILE% echo.  if %PAUSE_ERRORS%==1 pause exit  :failed echo APK setup creation FAILED.  if %PAUSE_ERRORS%==1 pause exit  :end I'm looking to clarify the differences/overlaps between the three and, if the terminology isn't accurate establish the correct terminology The drive is constantly filling up. You've hunted down all the spare and random files you can. grep'd for coredump files and even removed some of the un-needed backups...  What would be your next move.  The actual server in question has 10GB of website files and the OS shouldn't take any more than 10GB so how do you track down what's filling a 50GB (virtual) drive? Sum of a Hyper-geometric series. (NBHM 2011) How to find the sum of the following series $$\frac{1}{5} - \frac{1\cdot 4}{5\cdot 10} + \frac{1\cdot 4\cdot 7}{5\cdot 10\cdot 15} - \dots\,.?$$  I have no idea. I have written the general term and tested its convergence by Gauss' test for convergence, but they are neither the question nor the answer. I know that $$\gcd(a^b-1,a^c-1)=a^{\gcd(b,c)}-1$$ which can be seen by expanding both terms into geometric series. Is there any such simplification for $$\gcd(a^b+1,a^c+1)=?$$  EDIT: When working with polynomials instead of whole numbers, it is $$gcd(x^n+1,x^m+1)=\begin{cases}a^{\gcd(n,m)}+1 &amp; \nu_2(n) =\nu_2(m)\\1 &amp;\text{else}\end{cases}$$ But this gives only a lower bound for the whole-number case. How to avoid procrastination during the research phase of my PhD? How to determine a matching color code from a physical sample? I need to make a restaurant menu design that matches restaurant wallpapers. For some reason my client cannot get color codes used in wallpaper, but he provided me with a sample. (see below)    How can I get the correct color codes from the wallpaper? (someone mentioned that there are scanners for this purpose, but I could not find any information regarding that)  Let me know if this question should be in another site. How to prove that we are living in a 3+1D world? Is there any scientific experiment that can lead us to conclude we live in 3 spatial dimensions without the premise of the conception of limited dimensions?  Thank you all who helped in the improvement of this question (which was not clear at first).   EDIT:  I know that this can be a little philosophical, but it is also a scientific question.  Let's consider the scenario where the mankind was not ever able to see.  Let's also consider that this limitation could be surpassed thus not limiting us to reach a scientific and technological knowledge "similar" to what we have today.  Would this civilization of blind people reach the conclusion that they are living in a 3D spatial world?  Is the sense of touch enough to reach that conclusion? Is there any scientific experiment that can lead us to that conclusion without the premise of the conception of limited dimensions?  Would it be easier, harder, or just different to reach a conclusion predicted by the M-Theory? (please do not focus only on this last question) When compiling a beamer presentation and using the following \author command  \author{Name \\ \texttt{my.email@domain.com}}   I get the following hyperref warning in my logfile  Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref)                removing `\\' on input line 15.   Package hyperref Warning: Token not allowed in a PDF string (PDFDocEncoding): (hyperref)                removing `\new@ifnextchar' on input line 15.   I understand that this has to do with hyperref setting the PDF metadata, where the linebreak does’t make much sense and should be removed.  Trying to set  \hypersetup{pdfauthor={Name}}   does’t change the situation, seems like hyperref is still looking at the author command.  How do I get the desired display of author name with email and still keep hyperref happy? What does standard deviation tell us in non-normal distribution In a normal distribution, the  imparts standard deviation a lot of meaning, but what would standard deviation mean in a non-normal distribution (multimodal or skewed)? Would all data values still fall within 3 standard deviations? Do we have rules like the 68-95-99.7 one for non-normal distributions? How do I alter a language item link? I am looking to modify the default output of Drupal 8 of links in a certain situation, the language switcher. Ideally I want to be able to do that in a twig template and not have to produce a  that overwwrites that whole link generation system and add a BIG if for the cases I need to overwrite.  I think the output layout should be controllable in the twig template similar to  (this answer outputs the same url for every languages)  I am using bootstrap as a base theme so i'd like to use the bootstrap markup for my links (btn btn-primary, dropdown)  Here is my code I am trying to make links--language-block.html.twig  {%- if links|length == 2 -%}     {# show only alternate language button #}     {%- for key, item in links -%}         {%- if not item.attributes['#options']['set_active_class'] -%} {# &lt;--- this is always true!? #}             &lt;a href="{{ item.link['#url'] }}" class="btn btn-primary"&gt;{{ item.link['#title'] }}&lt;/a&gt; {# #url is always the same what ever the language it is pointing to! #}         {%- endif -%}     {%- endfor -%} {%- elseif links|length &gt; 2  -%}     {# show selected language in button and other languages in drop down #}     &lt;button class="btn btn-primary dropdown-toggle" type="button" data-toggle="dropdown"&gt;{{ selectedLanguage }}&lt;span class="caret"&gt;&lt;/span&gt;&lt;/button&gt;     &lt;ul class="dropdown-menu"&gt;         {% for key, item in links %}             {%- if not item.attributes['#options']['set_active_class'] -%} {# &lt;--- this is always true!? #}             &lt;li&gt;&lt;a href="{{ item.link['#url'] }}"&gt;{{ item.link['#title'] }}&lt;/a&gt;&lt;/li&gt;             {% endif %}         {% endfor %}     &lt;/ul&gt; {%- endif -%}   Anyone has an idea on how to do this? Let $G$ be a finite abelian group and let $\phi: G \rightarrow G$ be a group homomorphism. I am trying to show that there is a positive integer $n$ such that $G \cong \ker(\phi^{n}) \times \phi^{n}(G)$.  I know that since $G$ is abelian we have that $\ker(\phi^{k})$ and $\phi^{k}(G)$ are normal subgroups of $G$ for any $k$  I also suspect we have the following towers which stabilize at $m$ and $m'$  $$\phi(G) \unrhd \phi^{2}(G) \unrhd \cdots \unrhd \phi^{m}(G)=\phi^{m+1}(G)=\cdots$$  $$\ker(\phi) \unlhd \ker^{2}(\phi) \unlhd \cdots \unlhd \ker^{m'}(\phi)=\ker^{m'+1}(\phi)=\cdots$$  I know that if given a group of the form $HK$ where $H$ and $K$ are normal subgroups of $HK$ and $H\cap K=1$ then $H \times K \cong HK$. I want to apply this to this situation but I am unable to show how write $G$ as a product $HK$  Resolution One of the comments directed me to the following article:  The last part of which answers the question. Below is what is says:  Choose $n=\max(m,m')$, then we have for $x \in \ker^{n}(\phi) \cap \phi^{n}(x)$, this means that $x=\phi^{n}(y)$ for some $y \in G$. This gives:  $0=\phi^{n}(x)=\phi^{2n}(y)$ which means that $y \in \ker^{2n}\phi=\ker^{n}{\phi}$. and then we have that $0=x=\phi^{n}(y)$.  The answer below kindly points out that every element $x$ is contained in one of the cosets of $G/\ker^{n}(\phi)$, this means that $x=k+g$ for some $k \in \ker^{n}(\phi)$ and $g \in G$. We also have that $g=\phi^{n}(h)$ for some $h \in G$. Writing $x=k+\phi^{n}(h)$ essentially shows that $G=\ker(\phi^{n}) + \phi^{n}(G)$. Using the fact above, the claim follows. Definitionally, we have that  $$ \int_\gamma f\ \overline{dz} = \overline{\int_\gamma \overline{f}\ dz} $$  Now let $\int_\gamma f\ dz = w = x +yi$.  Question 1: Is it not the case that $\int_\gamma \overline{f}\ dz = \bar{w} = x - iy$?  Question 2:  Assuming that $\int_\gamma \overline{f}\ dz = \bar{w} = x - iy$, then is it not the case that  $$ \overline{\int_\gamma \overline{f}\ dz} = \overline{x - iy} = x+iy = \int_\gamma f\ dz? $$  Basically, I'm failing to see how $\overline{\int_\gamma \overline{f}\ dz}$ isn't just $\int_\gamma f\ dz$. Moving from SQL 2005 [SQL_Latin1_General_CP1_CI_AS] to 2008 - will I lose any features by using 'backwards compatibility' If parsing the output of ls is dangerous because it can break on some funky characters (spaces, \n, ... ), what's the best way to know the number of files in a directory?  I usualy rely on find to avoid this parsing, but similarly, find mydir | wc -l will break for the same reasons.  I'm working on Solaris right now, but I'm looking for a answer as portable across different unices and different shells as possible. Is there a way to turn option strict off for just a single line of code?    I'm doing some maintenance work and I need to "cheat" in just one place and I don't want to lower the standard for the entire file. HDMI to VGA convert I just bought a new HP Pavilion dv7t Quad Edition laptop that has HDMI out. I already have a couple of monitors that are VGA and I'd like to hook up one of these to dual monitors.  I've read some sketchy reviews of products that try to convert directly from HDMI (digital) to  VGA (analog), so I was wondering if someone has a good recommendation on how to hook these two together.