My teacher once said, for any positive number $\ n, $ $\ n^k - 1 $ would always have $\ n-1 $ as a factor for all positive odd values of $\ k $. Could anyone tell me the proof? I have written my approach below.  Assuming $n-1$ to be a factor.  $n^k - 1 = (n-1)x $  $n^k = (n-1)x + 1 $  $k = \lg( (n-1)x + 1 ) / \lg (n) $   If my approach is right could you tell me where should I go from here?  Edit :  clears my doubt.  Edit 2 : My question is "why for any positive number $\ n, $ $\ n^k + 1 $ would always have $\ n+1 $ as a factor for all positive odd values of $\ k $ ".
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$$+3n + 8$ is divisible by $9$ which is $7+3+8=18=9\cdot2$.  Are they just a coincidence or is there a theory behind?  Is it about modular arithmetic?
Square bracket notation for dimensions and units: usage and conventions If I have a double (234.004223), etc., I would like to round this to x significant digits in C#.  So far I can only find ways to round to x decimal places, but this simply removes the precision if there are any 0s in the number.  For example, 0.086 to one decimal place becomes 0.1, but I would like it to stay at 0.08. Incompatibility with mathtools and unicode-math in xelatex? I find unexpected behaviour in using mathtools in conjunction with unicode-math, which shows up if the 'range' command is used. Some incompatibilities have been noted before, so perhaps this is related. However, it only showed up for me in Texlive2016; previous versions worked as expected. In the following example I wish to draw calligraphic fonts from a different set than the main maths fonts, and some calligraphic symbols, and a regular italic bold face math, are missing.   \documentclass{article} \usepackage{mathtools}  \usepackage{fontspec} \usepackage{unicode-math} \unimathsetup{math-style=ISO,bold-style=ISO,sans-style=italic} \setmainfont{xits-regular.otf} \setmathfont{xits-math.otf} \setmathfont[range={\mathcal,\mathbfcal}]{latinmodern-math.otf} %\setmathfont[range={}]{xits-math.otf} % Try with and without this line % \begin{document} Test $a+b = \mathcal{C H G}C \mathscr{D}$ $$-\frac{a}{b} \mathcal{O} . $$ \begin{equation}     \mathrm{d} = \mathcal G(\varphi,x,y,z,t) ,  \end{equation} The symbol $\mathcal{G}$ is missing.  The following equation is wrong if I include the line  ``setmathfont[range=\{\}] \{xits-math.otf\}'' \begin{equation}     \mathup{d} = \symbf{u}(\varphi,x,y,z,t) ,  \end{equation} The glyph $\symbf{u}$ is missing. \end{document}   If mmathtools is called after unicode-math the problem does not arise, but this ordering is not recommend by the unicode-math developers.  My specific question is, is this a bug or is it something I am doing wrong? Consider the examples:    "My name is Ricky."   He said his name is Ricky.    The tense is not backshifted in #2 because his name is and will always be Ricky. Also consider:    Ricky says: "I live in Paris."   Ricky said that he lives in Paris.    The tense is not backshifted in #4 because he still lives in Paris.  For examples #2 and #4, it is possible to not backshift the tense because the reporting words are still true at the time of reporting, but what will you say about the following examples:        I asked what his name is.   I asked where he lives.      Here, is it correct for #5 and #6 to not backshift the tense?  If #5 and #6 are correct, then under what conditions is it correct to not backshift the verb like that? How can one change an entire document to use bold font? I see many posts in how to changes the font and font size, but none for changing the default font for a document to bold font. I believe this is called changing the font series. The approach in  (obtained by random Googling) looks like it should work, but doesn't do anything when I stick it in the preamble. Maybe I'm putting it in the wrong place.  \fontencoding{T1} \fontfamily{garamond} \fontseries{m} \fontshape{it} \fontsize{12}{15} \selectfont   There is also some discussion . Is it possible to query the content of the Time-based Workflow Queue from an Apex unit test to assess that a specific Workflow Rule has been executed and added a new item to the queue? I didn't see anything alike in the documentation, but wish it could be possible to improve testing and stability overtime.  Thanks I want to unwrap a label which has a transparent area Placing however at the start of a sentence vs placing it in the middle of a sentence? Basically, what's the difference between this:     However, it didn't mean that I didn't have the potential to become a   rock star.   and this?     It didn't mean, however, that I didn't have the potential to become a   rock star.   Or does it mean exactly the same thing? Please add a spell checker to the Markdown editor It would be nice to see a built-in spell checker into the main text-panel for posts. Every time I post here or Stack&nbsp;Overflow I quickly open up Gmail, paste my post there, and use its spell checker to make sure I didn't make any stupid mistakes. I know other services have these types of features, and it would be nice to see Stack&nbsp;Overflow have one as well.  Agree/disagree?  (BTW, I found two small mistakes for this short post.)  I know I made a comment about this on ChrisF's answer, but I thought that I should also mention it here in the question. I know there are many third-party tools to add spell checker functionality to Stack&nbsp;Overflow. But some (if not many) people out there do not have these tools. They are either at work (like me), or some other place where they using a machine that is not owned by them, or they aren't power users and just don't have these tools installed.  If Stack&nbsp;Exchange becomes used for other non-technical sites, then you will probably have to count on the fact that people are using  or something just as bad. That's why it would just be nice to have a built-in spelling tool. Again, it may not be worth it, but I think it would be interesting to discuss it at least. How is the derivative of $\mbox{Trace}\left\{ X^T A X B\right\}$ with respect to matrix $X$ equal to $AXB + A^TXB^T$?  \begin{align}   \nabla_X \ \textrm{Trace}\left\{ X^T A X B\right\} = AXB + A^TXB^T \end{align}  where $A$ and $B$ matrices are given. How to reconcile the cashless human/Federation society with human traders and merchants From a variety of sources, including , we know that the Federation (or at least the humans) don't use money. This raises the problem that humans frequently visit worlds where there is currency, and, in the case of DS9, operate a station that has currency frequently in use (for example, Federation personnel frequently attend Quark's bar). Additionally, we see human traders.   My question is, how do humans get "money" (credits, latinum, etc.)? How would a human even get the resources to purchase their own merchant ship or the resources to fill its cargo holds with merchandise? If Federation personnel can simply be given money for the asking, why haven't they poured so much money down Quark's throat that he chokes on it? Add CSS or JavaScript files to layout head from views or partial views Finding the limit $\lim \limits_{n \to \infty}\ (\cos \frac x 2 \cdot\cos \frac x 4\cdot \cos \frac x 8\cdots \cos \frac x {2^n}) $ Finite Index of Subgroup of Subgroup Prove the following: If $H$ is a subgroup of finite index in a group $G$, and $K$ is a subgroup of $G$ containing $H$, then $K$ is of finite index in $G$ and $[G:H] = [G:K][K:H]$.  So this is basically a bijective proof? The number of cosets of $H$ in $G$ equals the number of cosets of $K$ in $G$ times the number of cosets of $H$ in $K$ by the multiplication principle?   Reference: Fraleigh p. 103  Question 10.35 in A First Course in Abstract Algebra Why is the edit button disabled? How to prove Fibonacci sequence with matrices? Let $S$ be an arbitrary amount and define the relation $R \subseteq \mathcal{P}(S) \times \mathcal{P}(S)$ so that $(A,B) \in R$ if and only if $A \supseteq B$. Here $\mathcal{P}(S)$ is a spelling for the power set to $S$. Show that $(\mathcal{P}(S),R)$ is a partially ordered quantity. 18.04 ctrl+shift to change language Can I use Ctrl+Shift for changing languages? I can't set this hotkey for changing keyboard languages.  At this moment Alt+Shift switch languages, but I can't see this hotkey in keyboard hotkeys list or in   gsettings list-recursively  org.gnome.desktop.wm.keybindings | sort | more   output. won't instead of using' wouldn't' is possible? I have asked many questions about "would" both in ELL and ELU. But I still have a question about the use of "would". I have recently asked a question which @Jay also answered. He used "wouldn't" in the following sentences: 1. Here, you wouldn't say, "May I ask you what is your name?" Though we break this rule for questions sometimes 2. Well, someone might say that in informal speech, but you wouldn't write it or use it in formal speech. 3. "May I ask you what's your name?" is awkward; a fluent speaker wouldn't say that.  I had asked  question in ELL and I have received answers for it. But there is one thing that I'm still confused about.  Do the meanings of the sentences above change if we use 'won't' instead of using 'wouldn't' ? can we use 'won't' in the place of 'wouldn't' without changing the meanings of the sentences above? How critical are macOS Mojave compatibility requirements? As per Apple support document, :     1. Check compatibility      You can upgrade to macOS Mojave from OS X Mountain Lion or later on   any of the following Mac models. Your Mac also needs at least 2GB of   memory and 12.5GB of available storage space, or up to 18.5GB of   storage space when upgrading from OS X Yosemite or earlier.         MacBook introduced in early 2015 or later   MacBook Air introduced in mid 2012 or later   MacBook Pro introduced in mid 2012 or later   Mac mini introduced in late 2012 or later   iMac introduced in late 2012 or later   iMac Pro (all models)   Mac Pro introduced in late 2013, plus mid-2010 or mid-2012 models with a recommended Metal-capable graphics card.      I am currently running macOS High Sierra on my Early 2011 MacBook Pro.  What does Apple's requirements mean for my ability to install Mojave? Is it possible or worthwhile to do so?