How can we prove that if : $n+1|n!+1$ then $n$ is a prime number. My try: I try to do contrapositive: If $n$ is composite then $n+1 \nmid n!+1$ and then to get that if $n$ is composite so $n+1=ab$ when $2 \le a \le n $ so that $a|n+1$ but here I don't have idea how to continue, is this the good way?
... namely that if $n &gt; 1$ and $(n − 1)!\equiv−1\pmod{n}$, then $n$ is prime.  This is for a number theory class I'm in at Penn State. My idea is to follow accordingly, but I can't get it concretely..  Reason by contradiction, suppose that $n$ is not prime.   Then $n = ab$, where $1 &lt; a, b &lt; n$  In particular $(n-1)! = ai$ for some $i\in\Bbb{Z}$. $\\$ If $(n-1)! = nj - 1$ for some $j\in\Bbb{Z}$ then it follows that a probably doesn't divide $-1$?
Metrizable topological space $X$ with every admissible metric complete then $X$ is compact How to prove:      If $X$ is a metrizable topological space and every admissible metric on $X$ is complete then $X$ is compact.    I was trying with an idea of contradiction and thereby to construct an incomplete metric on $X$. But not able to construct. How can I get this 'Version: 0.1.0 - 94" value from the  tag in selenium webdriver? How can I get this Version: 0.1.0 - 94 value from the &lt;div&gt; tag in selenium webdriver? and here is my code:  &lt;div _ngcontent-qbk-c3="" style="text-align: center;"&gt;Version: 0.1.0 - 94&lt;/div&gt; why do academics write peer reviews? On Windows 7, is there a command prompt line that can start cmd as an administrator? On Linux, we can use  sudo bash   to run a shell as the superuser.  On Windows 7, we can use     Start menu -> (in search box, type) cmd -> Right click on search result and choose Run as Administrator   to run a command prompt as an administrator. Is there also a command prompt line that is like Linux's sudo or su so as to start a shell as an administrator? How to delay or slowly loop commands? I found 11 questions like this on gamingSE already, but none of them have proper and up-to-date answers. So I'm creating this Q&amp;A:  How do you either activate a command x seconds/minutes/hours/… in the future or every x seconds/minutes/… in a loop?  Repeating command blocks and ticking functions execute 20 times per second, redstone is  in many situations and most of the extremely fancy solutions, like falling blocks in cobwebs, have broken over time or are needlessly complicated. Every time I go to install a new app, I get the message "Insufficient storage", and an unable to install anything!  I am using Samsung GT19060 my internal memory is 8GB. I have not downloaded anything but anytime I try to download something it will tell me low memory space. Isomorphism $H_n(D^n, S^{n - 1};A)$ and $A$ The homomorphism $a \mapsto [[a \cdot f]]$ is supposed to be an isomorphism between $A$ and $H_n(D^n, S^{n - 1};A)$ for any homeomorphism $f: \Delta^n \to D^n$. I have proven that it is well defined and injective, but why is it surjective?  The group $H_n(D^n, S^{n - 1};A)$ is generated by elements of the form $a \cdot \sigma + b \cdot \tau + c \cdot \partial(\kappa)$ where $\sigma: \Delta^n \to D^n$, $\tau: \Delta^n \to S^{n - 1}$ and $\kappa: \Delta^{n + 1} \to D^{n + 1}$ and I do not see how an element of this form represents the same equivalence class of $[[m \cdot f]]$ for some $m \in A$. We can write $\sigma$ as $f f^{-1} \sigma$ to get $f$ in there, but I don't see that working out. Any hints? What is a reasonable depth and diameter for postholes dug in clay soil to support an 8' fence? What is a reasonable depth and diameter for postholes dug in clay soil where the frost line is no more than 12” and that will support an 8 foot tall solid wood fence that uses PostMaster metal posts?  I hope to have someone else install an 8 foot high PT wood fence.  There will be no light or air passing through or under the fence and it will be heavier than average, given the additional height and board weight.  One side will run for 135’.  The attached diagram provides more detail.  The PostMaster Installation Instructions describe a 30” deep, sloped, 10” diameter hole, filled with 6” of gravel plus 24” of concrete for a 6 foot high fence.  They caveat that the “exact diameter and depth will be determined by local weather and soil conditions.”  Got it.  Huntsville, AL; very clay soil.  Maximum wind gusts of 65 mph.  (This is not in the normal tornado path and is not intended to withstand a tornado.)  In reviewing the StackExchange posts on fencing and post holes I came across the concept that the diameter of the pier may be increased to compensate for a shallower post hole.  I would like to know the recommended depth and diameter without relying on the 2 buttress posts, which are my small overbuild idea.  We will be able to walk under them.  26 APR 18 update: I just found a Master Halco youtube video specifying 6-8" diameter holes dug 36" deep for an 8' board-on-board fence, such as I would like to have.   - I also spoke to a fence builder experienced using PostMaster posts who opined that the 10" diameter and 30 deep holes were more than adequate, but that the gravel was of no value in our climate.  So, 30" x 10" dia inches of concrete vs. 36" x 8" dia of concrete. I have an android phone to which i have plugged an earphones. So at the top of the phone, I get the headphone symbol which indicates that the earphone is connected (In other words, the circuit at the 3.5 mm jack is closed).   Then I cut the two earphones (transducers) from it, and still the headphone symbol shows. When I later cut this cable, below where it branches out, even then it shows circuit completion.   So my question is this:   How does the phone detect circuit completion at the 3.5 mm jack and thus trigger all sound and music to be directed through the 3.5mm jack? What is a good asymptotic for $f_n = f_{n-1}+\ln(f_{n-1})$? Given that $a$~$\mathcal{N}(\mu, \Sigma)$ and $\mu$~$\mathcal{N}(0, \sigma^2 I)$, Is $p(a|\Sigma, \sigma)=\int p(a|\mu, \Sigma)p(\mu|\sigma)d\mu$ again a Normal distribution? And how to estimate the mean and variance respectively? Thanks a lot! What is the difference between a linear and a circular polarizer? When I bought my polarizer, a friend told me that I should get a circular one, because the linear ones can mess with the autofocus. Is this true? What should each be used for? When closing a question as a duplicate there is a text box which says: "search, or enter a question link or numeric id". You can copy and paste a question link into here, but this fails if the URL starts with https:// instead of http://, with the message "Your search returned no matches; please try a different search". How do I right align controls in a StatusStrip? How to drive multiple LEDs from a Li-ion battery (with varying voltage)? I frequently need to do the following things in MySQL:   Create a non-root user Set that user's password Grant that user all privileges Make a database with the same name as the new user Allow usage only for users from the localhost.   Previously I did this using PHPmyadmin, but I would prefer doing it directly from BASH. Is there a CLI way to execute these steps? The dissolve tool (vector>geoprocessing tools>dissolve) renders some artifacts on shapes with high number of polygons. I’m using a  with 3222 polygons and I’m trying to dissolve it using the field “municipio”, but when the dissolve tool ends calculating the new shape has some artifacts     Does anyone had a similar problem? How to work around it?  I’ve also used v.dissolve from GRASS and the dissolve polygon from SAGA, but the problem persists. I’ve used the same shapefile on ArcGIS 9.3 with the dissolve tool. None artifact/error was produced.  Edit 1  I'm trying to use the v.clean tool as sugested by WhiteboxDev. However I'm getting this error:     An error has occured while executing Python code:      Traceback (most recent call last):   File   "C:/PROGRA~1/QGISCH~1/apps/qgis/./python/plugins\processing\gui\ProcessingToolbox.py",   line 181, in executeAlgorithm   File   "C:/PROGRA~1/QGISCH~1/apps/qgis/./python/plugins\processing\algs\grass7\Grass7Algorithm.py",   line 511, in checkBeforeOpeningParametersDialog   File   "C:/PROGRA~1/QGISCH~1/apps/qgis/./python/plugins\processing\algs\grass7\Grass7Utils.py",   line 361, in checkGrass7IsInstalled   File   "C:\PROGRA~1\QGISCH~1\apps\Python27\lib\ntpath.py", line 96, in join   TypeError: object of type 'NoneType' has no len()      Python version:   2.7.4 (default, Apr  6 2013, 19:54:46) [MSC v.1500 32 bit (Intel)]      QGIS version:   2.4.0-Chugiak Chugiak, 8fdd08a      Python path:   ['C:/PROGRA~1/QGISCH~1/apps/qgis/./python/plugins\processing',   'C:/PROGRA~1/QGISCH~1/apps/qgis/./python',   u'C:/Users/CERCOTRICHAS/.qgis2/python',   u'C:/Users/CERCOTRICHAS/.qgis2/python/plugins',   'C:/PROGRA~1/QGISCH~1/apps/qgis/./python/plugins',   'C:\PROGRA~1\QGISCH~1\bin\python27.zip',   'C:\PROGRA~1\QGISCH~1\apps\Python27\DLLs',   'C:\PROGRA~1\QGISCH~1\apps\Python27\lib',   'C:\PROGRA~1\QGISCH~1\apps\Python27\lib\plat-win',   'C:\PROGRA~1\QGISCH~1\apps\Python27\lib\lib-tk',   'C:\PROGRA~1\QGISCH~1\bin',   'C:\PROGRA~1\QGISCH~1\apps\Python27',   'C:\PROGRA~1\QGISCH~1\apps\Python27\lib\site-packages',   'C:\PROGRA~1\QGISCH~1\apps\Python27\lib\site-packages\GDAL-1.11.0-py2.7-win32.egg',   'C:\PROGRA~1\QGISCH~1\apps\Python27\lib\site-packages\PIL',   'C:\PROGRA~1\QGISCH~1\apps\Python27\lib\site-packages\jinja2-2.7.2-py2.7.egg',   'C:\PROGRA~1\QGISCH~1\apps\Python27\lib\site-packages\markupsafe-0.23-py2.7-win32.egg',   'C:\PROGRA~1\QGISCH~1\apps\Python27\lib\site-packages\python_dateutil-2.2-py2.7.egg',   'C:\PROGRA~1\QGISCH~1\apps\Python27\lib\site-packages\pytz-2014.2-py2.7.egg',   'C:\PROGRA~1\QGISCH~1\apps\Python27\lib\site-packages\win32',   'C:\PROGRA~1\QGISCH~1\apps\Python27\lib\site-packages\win32\lib',   'C:\PROGRA~1\QGISCH~1\apps\Python27\lib\site-packages\Pythonwin',   'C:\PROGRA~1\QGISCH~1\apps\Python27\lib\site-packages\Shapely-1.2.18-py2.7-win32.egg',   'C:\PROGRA~1\QGISCH~1\apps\Python27\lib\site-packages\six-1.6.1-py2.7.egg',   'C:\PROGRA~1\QGISCH~1\apps\Python27\lib\site-packages\wx-2.8-msw-unicode',   'C:\PROGRA~1\QGISCH~1\apps\qgis\python\plugins\fTools\tools',   'C:\Program Files\SpiderOak\shell_extension_lib\shared.zip']   Not entirely sure if this error is QGIS related... Any idea why I'm getting this error and how can I solve it? I am trying to work out how to get the value of table cell for each row using jQuery.  My table looks like this:  &lt;table id="mytable"&gt;   &lt;tr&gt;     &lt;th&gt;Customer Id&lt;/th&gt;     &lt;th&gt;Result&lt;/th&gt;   &lt;/tr&gt;   &lt;tr&gt;     &lt;td&gt;123&lt;/td&gt;     &lt;td&gt;&lt;/td&gt;   &lt;/tr&gt;   &lt;tr&gt;     &lt;td&gt;456&lt;/td&gt;     &lt;td&gt;&lt;/td&gt;   &lt;/tr&gt;   &lt;tr&gt;     &lt;td&gt;789&lt;/td&gt;     &lt;td&gt;&lt;/td&gt;   &lt;/tr&gt; &lt;/table&gt;   I basically want to loop through the table, and get the value of the Customer Id column for each row.  In the code below I have worked out that I need to do this to get it looping through each row, but I'm not sure how to get the value of the first cell in the row.  $('#mytable tr').each(function() {     var cutomerId =  } WITH ROLLUP WHERE x IS NULL Are there examples of third-(or higher)-order linear differential equations in physics or applied mathematics? The classical second-order linear ordinary differential equation is that named after Sturm and Liouville: formally, \begin{equation} (pu')'=ru. \end{equation} It arises naturally in many physical situations, for example through radial considerations of Schrödinger's equation. A higher-order "analogue" of it is the Orr$-$Sommerfeld equation, which, after relabelling the coefficients, may be written as \begin{equation} (\phi u'')''=\psi u. \end{equation} It also arises naturally, namely from certain simplifications applied to Navier$-$Stokes' (in)famous equation, and describes to great accuracy the cross-stream behaviour of channel fluid flow.  I have been thinking long and hard about whether there are additional higher-order linear differential equations that emerge naturally from our mathematical models of the world. It even seems pretty much all of the linear partial differential equations (heat, wave, Schrödinger, etc.) are of second order. Does anybody know of higher-order examples?