$(2^m - 1, 2^n - 1) = 2^d - 1$ where $d = (m,n)$ my work: I assumed $m = da$ , $n = db$ for $a,b \in \mathbb{Z}$. Now, $2^m - 1$ = $2^{da} - 1$ = $(2^d)^a - 1$ = $x^a - 1$ where $x = 2^d$.  similarly  $2^n - 1$ = $x^b - 1$ Now,  using $x^a - 1 = (x - 1)(x^{a-1} + x^{a - 2} + .... + x + 1)$ and $x^b - 1 = (x - 1)(x^{b - 1} + x^{b - 2} + ..... + x + 1)$      clearly $(x - 1)$ is a factor common in both but how to show that $(x^{a-1} + x^{a - 2} + .... + x + 1)$ and $(x^{b-1} + x^{b - 2} + .... + x + 1)$ can't have a common factor
Somewhere on Stack Exchange I saw the equation     $$\gcd(2^m-1,2^n-1)=2^{\gcd(m,n)}-1.$$   I had never seen this before, so I started trying to prove it. Without success...  Can anyone explain me (so actually prove) why this equation is true?  And can we say the same when replacing the '$2$' by any integer number '$a$'?
I have created a C# application which uses a SQL Server database. I have other computers connected to me and to each other in a workgroup. I have shared my C# application with others. When they open the application they get the error      A network related or instance-specific error occured while establishing a connection to SQL Server. the server was not found or was not accessible   But the application is working fine on my PC. The connection string I am using is   Data Source=ASHISHPC1\SQLEXPRESS;Initial Catalog=ACW;User ID=ash159;Password=ashish159   which is stored in a .config file.   The application is working fine on my PC. What must I do? I have enabled the TCP/IP in the server but the same error persists. Some change in connection string or something else? Error: Declaration of MyClass::start_lvl() should be compatible with that of Walker_Nav_Menu::start_lvl() I'm currently working on a WordPress site and whenever in the admin I go the Appearance > Menus page I get the following error:     ErrorException: Runtime Notice: Declaration of Walker_Nav_Menu_Edit::start_lvl() should be compatible with that of Walker_Nav_Menu::start_lvl() in wp-admin/includes/nav-menu.php line 203   It seems like some sort of PHP compatibility issue. I'm running PHP 5.3.10 on my local dev machine.  What would be the best way to remedy this problem? How to return to original .pdf presentation after open a .pdf linked file? I'm preparing a beamer presentation and I'm using hyperlinks for the first time; When I do something like:  \href{name_file_to_open.pdf}{here I put a link to a figure which is working well}   It works just fine and open the .pdf document; Problem is I can't return to my presentation, cause when I close the linked.pdf I close it all. (I try this with movie and indicating an area to the file open like \href[width=3cm,height=2cm] but then it doesn't work)  Can anyone please has any suggestion to this problem? Should moderators be allowed to decline flags on their own comments? How to put a variable in anchor tag Changing ChildOf constraint influence to 0, but keeping the position it had as a child I had a sword object rigged with a single bone, with a ChildOf constraint making it a child of a sheath (with an influence of 1). Throughout an animation i move the sheath, and the sword bone follows as expected.  However, i cant seem to lower the influence of the sheath to zero and still have the sword retain the position it was just at, and I am wondering how i should go about making this happen.  Thanks for your time. What is "angle of view" in photography? I read on the :     Focal-length determines the angle-of-view seen through a lens for a   given sensor-size. With a full-frame sensor a lens gives the same   angle-of-view as it would on a 35mm film camera. With a smaller   sensor, the angle-of-view becomes smaller. The crop-factor, also   called FLM, is the ratio representing the difference in equivalent   focal-lengths. So a 150mm on a full-frame DSLR such as the Nikon D700   gives the same angle-of-view as a 100mm on a D7000 since its FLM is   1.5X.      Short focal-lengths show a greater angle-of-view compared to longer   ones. [...]   In layman's terms, what is angle-of-view? Is it the same as focal length? If not, how is it different? How is it used? Why do I need to know about it? Although work and heat do the same thing (increase or decrease the internal energy of the system), There is still a fundamental difference between them. For example, The way in which entropy is defined is a very good way to differentiate between work and heat. But, why is there such a distinction between the two things? Is it the limitation of Newtonian mechanics that it never accounted for something like heat which could also change the energy of the system? Is the word "Thermodynamical work" or "Hidden work" suitable for heat? How does one install PlayOnLinux? How do I install PlayOnLinux on Ubuntu? What's amazing about these open world games is the freedom I have as a player. If I don't want to follow the main story line for a while, and just explore, I can do that, and the game won't break.  On the other hand I don't particularly like the feeling of having made a significant, bad choice in the game, among the lines of Yeah, that dialog option you selected hours ago? This amazing quest line is now off limits.  If I recall correctly, in Morrowind, I had to progress through a few different guilds' quests in the right order, otherwise it'd have stopped my progress in a few guilds.  As a Skyrim example, it doesn't appear that the very first choice made in Helgen at the very beginning has much significance. If someone told me It's not that significant, you still get to choose sides (or, rather not told me "This is important"), that'd help.  Another example: Based on a question I read here, I should probably make sure that I don't throw away any of the dragon priest masks and need to make sure to collect one during a specific quest. This would be relevant for this topic if I could easily miss it otherwise while doing that question.  This includes accidental problems possibly caused by bugs: It appears .  It doesn't matter too much which of the love triangle guys in Riverwood I support: That's not nearly significant enough.    Are there non-obvious, hidden decision points in the quests and story lines? When playing in a certain area, doing a certain quest, what do I need to make sure to do or not do, to not accidentally prevent me from getting or achieving something significant?  I still want to explore on my own, but want to be aware of which choices are important.  This is somewhat related to . Is there a way I can share the 3.42 GB download of Windows 8.1 from Windows Store? Windows 8.1 is out, and I would like to know if it is possible to share the Windows 8.1 download upgrade between different computers so that I don't have to download it over and over again?  This might be helpful for people who are on a metered connection. Store is downloading 3.42&nbsp;GB of data so it must be stored somewhere. Is there a way I can copy it to my other computers and start the setup so that I don't have to download it over and over again on each computer I own? Does iOS have a firewall? Many spells have "At higher levels" that provides for more targets or more damage.  For example Magic Missile can be cast as a second level spell and shoot 4 missiles instead of it's 3 as a first level spell.  Can a spell without that verbiage, such as Firestorm be cast in higher level slots for the same effect as the lower level spell? The model used to open the store is incompatible with the one used to create the store When I use the \tableofcontents macro in a document where the etoc package is loaded, latex pauses and reports an error. If I let the compilation continue a well formed pdf is generated.  Minimal example that can reproduce the error on my machine.  \listfiles \documentclass{article} \usepackage{etoc}  \begin{document} \tableofcontents‎‎ \end{document}   Output from pdflatex on the minimal example.  This is pdfTeX, Version 3.14159265-2.6-1.40.19 (TeX Live 2018) (preloaded format=pdflatex)  restricted \write18 enabled. entering extended mode (./mwe.tex LaTeX2e &lt;2018-04-01&gt; patch level 4 (/usr/local/texlive/2018/texmf-dist/tex/latex/base/article.cls Document Class: article 2014/09/29 v1.4h Standard LaTeX document class (/usr/local/texlive/2018/texmf-dist/tex/latex/base/size10.clo)) (/usr/local/texlive/2018/texmf-dist/tex/latex/etoc/etoc.sty (/usr/local/texlive/2018/texmf-dist/tex/latex/tools/multicol.sty)) (./mwe.aux)  ! Package inputenc Error: Unicode character ‎ (U+200E) (inputenc)                not set up for use with LaTeX.  See the inputenc package documentation for explanation. Type  H &lt;return&gt;  for immediate help.  ...                                                l.6 \tableofcontents‎                        ‎ ? R OK, entering \nonstopmode...  ! Package inputenc Error: Unicode character ‎ (U+200E) (inputenc)                not set up for use with LaTeX.  See the inputenc package documentation for explanation. Type  H &lt;return&gt;  for immediate help.  ...                                                l.6 \tableofcontents‎‎  [1{/usr/local/texlive/2018/texmf-var/fonts/map/pdftex/updmap/pdftex.map}] (./mwe.aux)   *File List*  article.cls    2014/09/29 v1.4h Standard LaTeX document class   size10.clo    2014/09/29 v1.4h Standard LaTeX file (size option)     etoc.sty    2018/02/23 v1.08n Completely customisable TOCs (JFB) multicol.sty    2018/04/20 v1.8s multicolumn formatting (FMi)  ***********   ) (see the transcript file for additional information)&lt;/usr/local/texlive/2018/te xmf-dist/fonts/type1/public/amsfonts/cm/cmbx12.pfb&gt;&lt;/usr/local/texlive/2018/tex mf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb&gt; Output written on mwe.pdf (1 page, 16281 bytes). SyncTeX written on mwe.synctex.gz. Transcript written on mwe.log. How to stop HTTPS requests for non-ssl-enabled virtual hosts from going to the first ssl-enabled virtualhost (Apache-SNI) I hope that title is clear.  How do I prevent HTTPS requests for non-ssl-enabled virtual hosts from going to the first ssl-enabled virtualhost (setup is Apache-SNI).  For example, using my abbreviated config below, requests for https://example.com (a non-ssl vhost) are being served by Apache at the ssl-enabled vhost https://example.org. I'd like to disable that behavior and possibly reply with the appropriate HTTP response code (unsure of what that is).  It may not even be possible, but I thought I'd ask.  # I actually have a SNI setup, but it's not demonstrated here. # I don't think it's relevant in this situation.  NameVirtualHost *:80 NameVirtualHost *:443  &lt;VirtualHost *:80&gt;     ServerName example.org &lt;/VirtualHost&gt;  &lt;VirtualHost *:443&gt;     ServerName example.org &lt;/VirtualHost&gt;  &lt;VirtualHost *:80&gt;     ServerName example.com &lt;/VirtualHost&gt;   EDIT: Maybe a mod_rewrite rule in the first ssl-vhost? How do i install drush for a drupal site which is on shared hosting? Uniform convergence of alternating series Looking for a simple proof that groups of order $2p$ are up to isomorphism $\mathbb{Z}_{2p}$ and $D_{p}$ . How to generate world files for raster X?