If $a,b \in \mathbb{Z}$ and odd, show $8 \mid (a^2-b^2)$.   Let $a=2k+1$ and $b=2j+1$. I tried to get $8\mid (a^2-b^2)$ in to some equivalent form involving congruences and I started with $$a^2\equiv b^2 \mod{8} \Rightarrow 4k^2+4k \equiv 4j^2+4j \mod{8}$$ $$\Rightarrow k^2+k-j^2-j=2m$$ for some $m \in \mathbb{Z}$ but I am not sure this is heading anywhere that I can tell.  Second attempt: Use Euler's Theorem and as $\gcd(a,8)=\gcd(b,8)=1$ and $\phi(8)=4$, $a^4 \equiv b^4 \equiv 1 \mod 8$ so $a^4-b^4\equiv 0 \mod{8}$.  I haven't gotten too much further are there any hints?
Prove: If $a$ and $b$ are odd integers, then $8\mid (a^2-b^2)$  So far I have: $$8\mid \left((2n+1)^2-(2m+1)^2\right)\Longleftrightarrow 8\mid \left(4n^2+4n+1-4m^2-4m-1\right) $$ Is this right so far? Where do I go from here? Step by step explanation please!
Can I visit the UK with an EU permanent residence card? I am non EU national, living in a Schengen country for past 5 years. I hold Indian passport and EU permanent residence card valid for 5 years.   Can i go to UK for few days to visit friends without a visa?  I am aware that UK is not in Schengen, and Schengen visas and residence permits issued by Schengen country is not valid for United Kingdom? But what about the PR card issued by a Schengen country?  I am also aware that EU citizens carrying EU passport have no restrictions to visit UK. I always thought the book was Manifest Destiny but its not. The story that sticks in my head was a human in a cage over a pit? Aliens walk in and discuss how dangerous humans were. He eventually escapes. Man I miss this book. wish I wouldve written down every book I ever read. Ugh How do I mention my own products in answers? Twice now I've seen questions on Stack Overflow where one of my company's products might provide an answer to the question.  One was a slightly vague connection about object-relational mapping, but the other was a specific question about workshop/event management where we have a product which is almost an exact match for the questioner's needs.  Of course the questioner should take a pinch of salt with anything I say about it, but it might be useful to have the product brought to his attention as one possible solution.  What are the rules and guidelines about answering with something I worked on? How do I install/update drivers on Debian 9? I just installed Debian 9 on my laptop, however Wifi isn't working and I'm not sure if my graphic card is either. I'm sure that it's just a lack of drivers, but I've never actually had to update drivers on Linux before. So how do I do that? "It is not possible to check out a single file. The finest level of checkouts you can do is at the directory level."  How do I get around this issue when using Subversion?  We have this folder in Subversion where we keep all our images. I just want to check out one file (image) from that. This folder is really big and has ton of other stuff which I don't need now. Is there such a thing as a restore point in Ubuntu? Windows has this. Basically what I want is to record the state of the system either manually or periodically and then restore to that point if required. For example, before upgrading the distribution, or before installing a certain package.  What would be the easiest way to do this? How to obtain and use the .bbl file in my tex document for ArXiv submission? I am submitting my first article to ArXiv and in the instructions they specify I have to include the .bbl file. Their instructions are:  "We do not run BibTeX in the auto-TeXing procedure. If you use it, include in your submission the .bbl file that BibTeX produces on your own machine; otherwise your references will not come out correctly. We do not run BibTeX because the .bib database files can be quite large, and the only thing necessary to resolve the references for a given paper is the .bbl file. The name of the .bbl file must match the name of the main .tex file for the system to process the references correctly."  My knowledge of LaTeX is basic, and I have not heard of .bbl file before. I am using TeXmaker. I have my tex file and my bib file. So what do I exactly do to obtain the .bbf file correctly and submit the article?  Thanks a lot! How to make a VGA monitor display a white screen at all time? How to draw circle in html page? I tried to install crossover and after downloading the .deb from their website, I followed the steps as indicated when opening it with Ubuntu Software Centre:     On 64-bit flavors of Debian and some Ubuntu variants, first open a   terminal and run sudo dpkg --add-architecture i386 ; sudo apt-get update   Since I have Ubuntu 16.04 LTS 64 bit I perform the command.  But when copying and pasting that command into my terminal, it stopped at Fetched 535kb in 7sec and stalled there for about 20 minutes. I ended up  closing it and tried to run both commands separately but after doing the sudo apt-get update command it also stalls again after "Fetched XXKb in XXsec."  I restarted my computer and decided to run the software updater and it stays stuck at roughly 75% for hours. A real $2 \times 2 $ matrix $M$ such that $$M^2 =  \begin{pmatrix} -1&amp;0 \\ 0&amp;-1-\epsilon  \\   \end{pmatrix}$$ (a) exists for all $\epsilon &gt; 0$. (b) does not exist for any $\epsilon &gt; 0$. (c) exists for some $\epsilon &gt; 0$. (d) None of the above is true.  Attempt: I couldn't think of any theory base to prove / disprove the existence of such a matrix. Could someone please give me a hint on how to go about this problem? Thank you very much for your help in this regard. Search among posts you've voted for Related posts:    These posts are different from mine in that they merely ask for the &quot;votes&quot; tab back or ask if it is at all possible to get such information, while I propose how to implement this as a search feature. Background I recently found myself facing a problem very similar to one that I had solved before. I remembered that I found the solution on SO and I remembered upvoting the post containing the solution. I wasn't able to find that particular post by just searching with relevant keywords. It would have been nice if I could search only among the questions I have voted on. I searched on meta and found that there had been a tab for looking at what you've voted on, but that it was removed due to lack of interest. This is understandable, since just listing the hundreds/thousands of posts you've voted on isn't particularly useful. Proposal What I want is to be able to search for posts I've voted on similarly to the way I'm able to search on my posts within a specific tag. For instance: user:310121 [iphone] apple  This will return all my posts tagged iPhone, where I mention Apple. This is brilliant. This is just how I want search to work. Now what I want to do is to extend this to include votes. The exact syntax isn't relevant, but I had to make one up to show you what I mean: To show all posts you've voted on: user:310121 {votes}  To show all posts you've upvoted: user:310121 {upvotes}  To show all posts you've downvoted: user:310121 {downvotes}  You could probably omit the the user part, since you're only allowed to see your own votes. (Unless it's desirable to be able to look at other peoples votes. Maybe you should be allowed to do this if you have enough rep? Well, that's another discussion.) I guess that even if you're allowed to see other peoples votes it would make sense to display your own votes if the user part is omitted. So in my case I could have done the following search and found the post I was looking for: {upvotes} keyword  Lastly I just want to suggest that clicking on your vote count will work just like clicking on a tag. If I click on a tag in my profile the following search will be done: user:310121 [tag]  So if I click on my upvote count the following search should be made: user:310121 {upvotes}  The same goes for downvotes: user:310121 {downvotes}  This would be a very useful feature and it wouldn't be intrusive in any way. The users that don't care for this feature might never know that it's there. And just for the record the favorite system doesn't replace this feature. Accidentally ejected External drive while downloading file, now having problems reading How to send data to a serial port and see any answer? On Linux, I want to send a command string (i.e. some data) to a serial port (containing control characters), and listen to the response (which also usually might contain control characters).   How can I do this as simplest as possible on Linux? An example is appreciated! How to create quizzes with macqtex? Generalized RBF Kernels There is the notion of Generalized RBF Kernels, for example in "Towards Optimal Bag-of-Features for Object Categorization and Semantic Video Retrieval" from Jiang (1) or in formula (2.72) in  (2).  They define: Gen. RBF Kernel: $e^{-pd(x,y)}$, where (1) says d can be any distance function and (2) requires it to be a metric.  I now have been wondering for a long time: Is a function defined by $e^{-pd(x,y)}$ where d is a metric always a Mercer's Kernel? I think in (2) it sounds like it, but if it were true, not many people know of it, since they are still looking to prove that $e^{-EMD(x,y)}$ is a kernel, where EMD = Earth Mover's Distance, which is a metric.  If its not a Mercer's Kernel - do you know of any counter example? What is the difference between an "API," a "framework," an "IDE," and an "engine?" What hardware does Ubuntu Touch support? I just saw Death in Paradise Season 5 Episode 3, which features a photographer using a DSLR to shoot a fashion runway performance.  He had a flash that was a ring around the lens.   I thought that was for macro shots only, where the extreme close-up has trouble with normal flash placement.  For normal photos you want the flash far from the lens, like on a boom on a separate handle framing the camera.  I've seen some recent trendiness in having ring-shaped catch lights, but I supposed those would be positioned in the normal manner: off-angle from the lens.  I suppose if an on-camera had any effect at all it would be for some subtle fill flash.  Is this a realistic thing to see from a pro? Or is it just a cool-looking prop that doesn't really fit the situation? How to determine whether a unique factorization domain is a principal ideal domain? Could someone please provide an example of a unique factorization domain that is not a principal ideal domain?  Furthermore, is there some way to determine whether a UFD is a PID?