Let $p$ be a prime $\geq 7$.  $p^2-1=(p+1)(p-1)$. Both $(p-1)$ and $(p+1)$ are divisible by $2$ and they are two consecutive even numbers. So, either one of the two must be divisible by $4$. So, $8 \mid (p^2-1)$. Again, by Euler's theorem, $p^2 \equiv 1 \pmod3$. Thereby, $24 \mid (p^2-1)$, for all prime $p \geq 7$.    To prove that $24$ is the greatest common divisor:  For $p=7$, $p^2-1=48$. So, the only other possible $\gcd$ is going to be $48$. But for $p=11$, $p^2-1=120$ and $48 \nmid 120$.  Therefore $\gcd\{p^2-1: p $  is a prime $ \geq 7 \}=24$.  Is this correct? Kindly verify.
Let $p&gt;3$ be a prime. Prove that $24 \mid p^2-1$.   I know this is very basic and old hat to many, but I love this question and I am interested in seeing whether there are any proofs beyond the two I already know.
IList vs IEnumerable for Collections on Entities When I have entities in my domain with lists of things, should they be exposed as ILists or IEnumerables? E.g. Order has a bunch of OrderLines. For a graph $G$, let $W$ be the (random) vertex occupied at the ﬁrst time the random walk has visited every vertex. That is, $W$ is the last new vertex to be visited by the random walk. Prove the following remarkable fact:      For the   random walk on an $n$-cycle, $W$ is uniformly distributed over all vertices different   from the starting vertex. How to select the rows with maximum values in each group with dplyr? What's the best way to draw UML class diagrams in LaTeX? Is any UFD also a PID? I have recently started using blender and one of the models I'm working on has dark shadows along edges. I started this model from scratch. I can't remember them being there when I first started the model, but I had trouble joining areas together and think I've pressed something I shouldn't have. However, there are so many options I'm not sure what I did.  Here is an image of what I mean   The one on the left has the shadow. As a test I started a new model, which is on the right in the image, and as you can see it doesn't have any shadows.  Also, occluded vertices partially show through the model. Occlusion doesn't work properly.   In orthogonal view small darkened triangles appear along the the edges.    I don't really want to start the model again from scratch so any ideas on how to remove this shadow would be great. Thanks. Identical coefficients estimated in Poisson vs Quasi-Poisson model I am using MikTex 2.9 and TeXstudio for my TeX preparation. I would like to set the path to my figures folder, which contains all the figures, plots etc. required for my work.   I used \graphicspath to set the directory, but it gives me an error message:  \documentclass[12pt]{article}  \usepackage{graphicx}  \graphicspath{D:/LATEX/Reports@IIT/figures}  \begin{document} \includegraphics[width=4.5cm,height=4cm,angle=0]{logo.jpg}  \end{document}      Error: Package pdftex.def Error: File `logo.jpg' not found   However, when I prepend the filenames of the images with the same path in the \includegraphics command, it works just fine:  \documentclass[12pt]{article}  \usepackage{graphicx}  \begin{document}  \includegraphics[width=4.5cm,height=4cm,angle=0]{D:/LATEX/Reports@IIT/figures/logo.jpg}  \end{document}   Why is \graphicspath not setting the path? jQuery - regexp selecting and removeClass()? Have a look at   One of the bolds works, and the other one just turns up as ** in the actual post.  This is the string that isn't working: **x**ml n**ib**  Which renders as: xml nib I would like to replace the usual \hat in maths with a larger one, but I do not want it to stretch. The size I want is "the smallest \widehat" (as in $\widehat{.}$). Is there a way to put that one above a wide character like M ($\widehat{M}$ would stretch and give be an even wider hat).  I guess I have a lack of understanding for the mechanism behind accents that extend, so feel free to explain. How to monitor battery condition and pop-up notification? Essentially, I would like  into a working answer.   I know how to extract the battery percentage from :   upower -i $(upower -e | grep BAT) | grep --color=never -E percentage|xargs|cut -d' ' -f2|sed s/%//   And how to pop-up a basic notification:   notify-send "battery low"   But how can I set up a (bash?) script to permanently monitor the output and send notification as per this pseudo-code:   if battery_status &lt; 10% then notify-send "battery low" and put my system to suspended state sudo pm-suspend For what powers $k$ is the polynomial $n^k-1$ divisible by $(n-1)^2$? Executing cmd.exe commands from Java Consider the brief passage:     "I love your work, but calling you 'the artist' {1} just doesn't seem to be   cutting it anymore. What shall I call you?"      "OK. Call me George." {2}   So my instincts tell me to put {1} in single quotes as opposed to no punctuation (or perhaps even italics) but to do the opposite for {2}. Is this right? Why?  Let me expand this one further iteration:     "Wow, that's a great name, especially with this crowd. Maybe for your next book    you should call yourself Ringo."{3} $a\equiv\bar a\!\pmod{\!kn}\Rightarrow a\equiv\bar a\!\pmod{\! n};\ $ $(a\bmod kn)\bmod n=a\bmod n.\ $ Congruences persist mod factors of the modulus I noticed relation between modulo operation and number which is power of two  Example I have to calculate $ 3431242341 \mod 2^5 $, which is $ 5 $ but it is equivalent to  $ ( 3431242341 \mod 2^9 ) \mod 2^5 $   I tried many examples and it seems to be true in general, and I am not sure if it is a coincidence or true in general that I can use first modulo operation ( greater number) and the result will be the same. I am trying to install Linux headers for Kali Linux on my machine and I have tried every possible solution on the internet but it always show "Unable to locate packages "  root@kali:/usr/sbin# apt-get install linux-headers-4.6.0-kali1-amd64 Reading package lists... Done Building dependency tree        Reading state information... Done E: Unable to locate package linux-headers-4.6.0-kali1-amd64 E: Couldn't find any package by glob 'linux-headers-4.6.0-kali1-amd64' E: Couldn't find any package by regex 'linux-headers-4.6.0-kali1-amd64'   Here is my sources.list file :  # Regular Repositories deb http://http.kali.org/kali sana main non-free contrib deb http://security.kali.org/kali-security sana/updates main contrib non-free # Source repositories deb-src http://http.kali.org/kali sana main non-free contrib deb-src http://security.kali.org/kali-security sana/updates main contrib non-free   Uname -a output :   root@kali:/usr/sbin# uname -a Linux kali 4.6.0-kali1-amd64 #1 SMP Debian 4.6.4-1kali1 (2016-07-21) x86_64 GNU/Linux Display command in xterm titlebar My Bash Prompt is currently setting the xterm titlebar using the following sequence:  PS1='\033]0;\u@\h&gt;\w\007'   Is there an easy way to display the current command in the titlebar. For example, if I am tailing a file using tail -f foo.log, I want my titlebar to say tail -f foo.log. How to automatically rotate images based on EXIF data? What is the difference between buy and purchase