I need to prove that $a^n - 1$ is a composite number when $a&gt;2$ and $n\ge2$ ($a,n$ are integers). I have tried a few things, but couldn't make it.
I did some mathematical induction problems on divisibility   $9^n$ $-$ $2^n$ is divisible by 7. $4^n$ $-$ $1$ is divisible by 3. $9^n$ $-$ $4^n$ is divisible by 5.   Can these be generalized as $a^n$ $-$ $b^n$$ = (a-b)N$, where N is an integer? But why is $a^n$ $-$ $b^n$$ = (a-b)N$ ?  I also see that $6^n$ $- 5n + 4$ is divisible by $5$ which is $6-5+4$ and $7^n$$+3n + 8$ is divisible by $9$ which is $7+3+8=18=9\cdot2$.  Are they just a coincidence or is there a theory behind?  Is it about modular arithmetic?
PHP script not executing on Apache server Proving that if $A^n = 0$, then $I - A$ is invertible and $(I - A)^{-1} = I + A + \cdots + A^{n-1}$ OSX (Yosemite) Make 2 sided printing the default for *all* applications I'd like to make double-sided printing the default for all applications.  This  tells how to set it for one application, but by experimentation it only affects that application, not others.  I Googled, looked in the system prefs, looked on this site - no luck.  Is there a setting somewhere? Why are these black dots / spots appearing in my pictures? How to write a good PhD literature review? What ticket reservation do I need for a Schengen visa? I have a visa appointment this Friday 19 at the Italian Consulate in Philadelphia and I have a question about the flight reservation requirement: According to the Visa requirements "Confirmed flight reservation for a round trip showing complete itinerary."  Does this mean I have to reserve the tickets, pay and print the itinerary?  In this case if my visa gets denied then I will lose the money I paid or can I just show the planned itinerary (written not booked)? in this case the travel agency would not give me the Travel Insurance coverage until I book and pay for the reservation.   I am a US green card holder and applying from North Carolina. $X=(-\infty,0]\cup\left\{{1\over n}:n\in\mathbb N\right\}$ with subspace topology Then $X=(-\infty,0]\cup\left\{{1\over n}:n\in\mathbb N\right\}$ with subspace topology. Then   $0$ is an isolated point $(-2,0]$ is an open set $0$ is a limit point of the subset $\left\{{1\over n}:n\in\mathbb N\right\}$ $(-2,0)$ is open set.   $0$ is not an isolated point of $X$ as every nbd of $0$ contains a point of the form $1\over n$ so  $3$ is true and $1$ is false, $2$ is false as $0$ is not interior point, $4$ is true as every point is interior point. Am i right? Display the original, unedited post on the post editing page I edit people's posts occasionally on ServerFault when the grammar/punctuation/coherence goes below a threshold that my OCD can handle. However, on some of the thornier posts I would like to be able to compare my edits to the original text, especially when doing a final proof-reading of my edits.  At the moment, my only options are to either copy the original text into a text editor or copy a duplicate of the text into the editing field. From there I can compare my edits to the original text and make sure I'm preserving the original meaning as I attempt to untangle less than optimal communication.  This request is somewhat similar to this Meta post: "" Another combinatorics problem: $\sum\limits_{k = 0}^n (-1)^k \binom{2n-k}k2^{2n-2k}=2n+1$ I am dealing with problem 10F in the textbook A Course in Combinatorics by J. H. van Lint and R. M. Wilson.   Problem 10F: Prove directly the equality  $\sum\limits_{k = 0}^n {{{\left( { - 1} \right)}^k}\left( {\begin{array}{*{20}{c}} {2n - k}\\k\end{array}} \right){2^{2n - 2k}} = } 2n + 1$.  Sketch of the proof: Evaluate  $\sum\limits_{n = 0}^\infty  {\sum\limits_{k = 0}^n {{{\left( { - 1} \right)}^k}\left( {\begin{array}{*{20}{c}}{2n - k}\\k\end{array}} \right){2^{2n - 2k}}{x^{2n}}} } $. To do this, use Hint 1. Then, use Hint 2.  Hint 1: $\sum\limits_{j = 0}^\infty  {\left( {\begin{array}{*{20}{c}}{a + j}\\j \end{array}} \right){x^j} = {{\left( {1 - x} \right)}^{ - a - 1}}} ,\,\,a \in {\mathbb{Z}^ + }$.  Hint 2:  ${\sum\limits_{i = 0}^\infty  {\left( {2i + 1} \right){x^{2i}} = \left( {\frac{x}{{1 - {x^2}}}} \right)} ^\prime }$.   Attempt: It is enough to show that  $\sum\limits_{n = 0}^\infty  {\sum\limits_{k = 0}^n {{{\left( { - 1} \right)}^k}\left( {\begin{array}{*{20}{c}} {2n - k}\\ k \end{array}} \right){2^{2n - 2k}}{x^{2n}}} }  = {\left( {\frac{x}{{1 - {x^2}}}} \right)^\prime }$. But I can not use Hint 1. How to disable Alt + ` (backtick/grave) What is the simplest SQL Query to find the second largest value? What is the simplest SQL query to find the second largest integer value in a specific column?   There are maybe duplicate values in the column. Show that any prime divisor of $x^4+x^3+x^2+x+1$, with $x\in\mathbb{N}$, is $5$ or $1$ mod $5$ We can write the "polynomial" as follows: $$x^4+x^3+x^2+x+1=\frac{x^5-1}{x-1}.$$ For even $x=2y$, we have that $x^5-1=(2y)^5-1=32y^5-1\equiv1$ mod $5$.  For odd $x=2y+1$, we have that $(2y+1)^5-1\equiv_532y^5\equiv2$ mod $5$.  I find it hard to believe my computations above. I guess this makes me rather clueless. I was hoping there was some way out with Legendre symbols. Any help is appreciated. I have installed nodeJS on Ubuntu with following code   sudo apt-get install nodejs   Since I am a new user for ubuntu I also ran this code too  sudo apt-get install npm   Now when I type    nodejs --version   It shows   v0.6.19   I checked and saw latest nodeJS version is 0.10.26   How can I update my version of nodeJS to 0.10.26?  I tried with    sudo apt-get install &lt;packagename&gt;  sudo apt-get install --only-upgrade &lt;packagename&gt;   but no luck. Can we calculate the standard error of prediction just based on simple linear regression output? The standard error of prediction in simple linear regression is $\hat\sigma\sqrt{1/n+(x_j-\bar{x})^2/\Sigma{(x_i-\bar{x})^2}}$.  My question is to calculate the standard error of prediction for $pop=1029$ just based on the following regression output. I can get all except $\bar{x}$. And I also know how to calculate the approximate standard error of prediction based on the standard errors of intercept and coefficient of $pop$, ignoring their correlation. How to select an element inside "this" in jQuery? How can I represent an 'Enum' in Python? I'm mainly a C# developer, but I'm currently working on a project in Python.  How can I represent the equivalent of an Enum in Python? Gtk skip_taskbar_hint (seen in Glade-ui-Designer) equivalent in Qt 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! Where does the term "fun fact" originate?-- namely, not with the compositional meaning but rather with the idiomatic usage to introduce some sort of unusual, esoteric, absurd or otherwise "controversial" fact? Can I change my Ubuntu Desktop into a different flavour (like Kubuntu)? Can I convert Ubuntu into another form of Ubuntu like Kubuntu or Ubuntu Studio and the system recognize itself as the new form?  UPDATE: After getting an answer and talking with people, I managed to gain a lot of information on this topic. I then wrote an article about converting between Ubuntu systems - What is the command to truncate a SQL Server log file? I need to empty an LDF file before sending to a colleague.  How do I force SQL Server to truncate the log?