If $m &gt; n$ and $a,m,n$ are positive, with $m$ not equal to $n$, find the  greatest common divisor of $2^{2^m}+1, 2^{2^n}+1$. Please solve this problem using Euclid's algorithm.  I tried to use Euclid's algorithm by first dividing $2^{2^m}+1$ by $2^{2^n}+1$, but I got stuck in the 2nd step while dividing $2^{2^n}+1$ by the difference of $2^{2^m}+1$ and $2^{2^n}+1$.
I've been working with the Fermat numbers recently, but this problem has really tripped me up. If the Fermat theorem is set as $f_a=2^{2^a}+1$, then how can we say that for an integer $b&lt;a$, the $\gcd(f_b,f_a)=1$?
I've been using mac for years, but prefer to do software development on windows. I'm really used to is using the folowing shortcuts on windows very much:  &lt;CTRL&gt;  + INSERT: Copy &lt;SHIFT&gt; + INSERT: Paste &lt;SHIFT&gt; + DELETE: Cut   Of late, I've taken to doing even my development work on the Mac. However, as a matter of habit, I'm missing these shortcuts very much. Actually, I find these a lot more convenient than COMMAND + (C/V/X) combo.  Is there a way to reassign these three shortcuts at least?   I've look all over the internet and in the "Keyboards" system preferences but I did not find anything.  PS: If you're wondering where I find the INSERT keys on a mac book pro, I use a wireless keyboard. How can I disable Ctrl+Q for all applications (system wide)? A lot of applications like Firefox, Chrome etc, get closed when I accidentally press Ctrl+Q instead of Ctrl+W because of the proximity of the Q and W keys on the keyboard. Is there a way this shortcut can be removed or reset on a system wide basis? How does the bounty system work? It's possible to offer and award bounties of reputation on questions.   What is a bounty? How can I find questions that have a bounty attached? How do I start a bounty? When can I start a bounty? Why is the system requiring me to choose a bounty amount that's higher than the minimum? Can I use Markdown formatting in the bounty remarks? Can I edit the remark? How long is the bounty period? How do I award a bounty? Can I award a bounty to my own answer? Can I award a bounty to an old answer? If I offer a bounty on someone else's question, can I award the bounty to their answer? After awarding the bounty, can I remove it or move it to another answer at a later time? What happens if there's no answer after the bounty period? What happens if I feel my question is still unanswered? What is automatic awarding? Are bounty awards exempt from the 200 points/day reputation limit? How does Community Wiki mode affect bounties? How does user deletion or post dissociation affect bounties? Can I award a bounty to an answer from a deleted user or one where the author was dissociated? Can I offer a second bounty after the first one has expired? Can I raise my bounty? What happens if a bounty question is closed or deleted? Why can't I vote to close or migrate a bounty question? What happens if I delete my account while I have an active bounty? Do I keep an awarded bounty when I delete my answer? What happens if a question where I awarded a bounty later gets deleted or migrated? Can I cancel my bounty? Can I affect / divide / share part of my bounty to two (or more) users? How many outstanding bounties can a single user have?      See also:  in the Help Center When can we speak of collinearity In linear models we need to check if a relationship exists among the explanatory variables. If they correlate too much then there is collinearity (i.e., the variables partly explain each other). I am currently just looking at the pairwise correlation between each of the explanatory variables.  Question 1: What classifies as too much correlation? For example, is a Pearson correlation of 0.5 too much?  Question 2: Can we fully determine whether there is collinearity between two variables based on the correlation coefficient or does it depend on other factors?  Question 3: Does a graphical check of the scatterplot of the two variables add anything to what the correlation coefficient indicates? What are your tips for improving overall system performance on ubuntu? Inspired by this  I realized that some default settings may be rather conservative on Ubuntu and that it's possible to tweak it with little or no risk if you wish to make it faster.  This is not meant to be application specific (e.g. make firefox load pages faster), but system wide.  Preferably 1 tip per answer, with enough detail for people to implement it.  A couple of mine would be:   Install  (via Software Center or sudo apt-get install preload); Change  value - "which controls the degree to which the kernel prefers to swap when it tries to free memory";   What are yours?  PS: Since this is not intended to have a unique answer but rather, several useful tips, I'm making this community wiki out-of-the-box. Cannot create PostGIS extension with Ubuntu 18.04 Western Digital drive making beeping noise How to correctly add a path to PATH? I'm wondering where a new path has to be added to the PATH environment variable. I know this can be accomplished by editing .bashrc (for example), but it's not clear how to do this.  This way:  export PATH=~/opt/bin:$PATH   or this?    export PATH=$PATH:~/opt/bin Can Ubuntu 32-bit access 8GB RAM? I need to use a large software package (Xojo) that requires the 32-bit architecture to run. Try as I might, I can't seem to install the 32-bit libraries on my 64-bit installation of 15.04.  I don't have a problem running a 32-bit version of Ubuntu (I'm only using this computer for Xojo) but the computer has 8GB of RAM and I'd like to be able to use it all. Is there a way to utilise all 8GB of RAM with a 32-bit Ubuntu install? Does Ubuntu do this out of the box? What is Attribute Based Encryption? I would like to place a lower limit under two consecutive \sums, somewhere in between. So I have \sum_{j=1}^{\infty}\sum_{k=1}^{\infty} and would like to place j&gt;k in the lower limit between the two sums. I believe the multiple integral operator has this kind of functionality. I'm trying to condense the space around examples in gb4e, with the goal of removing unnecessary whitespace around examples in an abstract. Here's what gb4e gives me now:    What I'd like is for the examples to essentially look like part of the running text, as follows:    In sum, I'd like to remove the left indent before the top level item numbers, remove the vertical space between examples, and remove the vertical space above and below the exe environment.  I think \vspace commands might work here, but it would get very tedious to do this every time, and I'd also like to do it the 'right' way.  (I also think linguex might be able to do this without too much fuss, but I'm used to using gb4e so I'd prefer to stick with it if there's a way to do this.)  MWE:  \documentclass{article}  \usepackage{gb4e} \usepackage{lipsum}  \begin{document}     \lipsum[1]     \begin{exe}         \ex Blah \begin{xlist}             \ex Blah             \ex Blah         \end{xlist}         \ex Blah         \ex Blah     \end{exe}     \lipsum[1] \end{document} Rollback Committed transaction in SQL Server Is it possible to rollback a committed transaction?  I have deleted all the data in a table. The delete statement is inside Begin Transaction and I have committed the transaction with a commit transaction command. Now is it possible to rollback and get the data back? Does the universe have a center? Update pgfmath macro in foreach loop I need to update a variable defined in a pgfmathsetmacro command inside a foreach loop. Apparently, the var is only updated when it is used after the update command. Here is a MWE:  \begin{tikzpicture} \draw [help lines] (0,0) grid (3,3); \pgfmathsetmacro{\a}{1} \foreach \b in {2,3} {     \node [blue] at (\a,0) {\a};     \pgfmathsetmacro{\a}{\b}     \node [red] at (0,\a) {\a}; } \end{tikzpicture}   The red nodes are printed at updated positions and the value printed is also updated; the blue nodes are not: they keep their initial value (1) and are located at the initial position (0,1).  Why is this? Thanks José How to store date/time and timestamps in UTC time zone with JPA and Hibernate This question is similar, but not identical to:   Whereas the aforementioned question is about whether the hand’s movement is measured from the hand’s point of view or from the caster’s, this question revolves around the total distance the hand can travel to reach its destination.  To illustrate, let me give an example: Two narrow stone pillars stand next to each other, 10 feet apart. They are several kilometers high and on top of each pillar stands a wizard. One wizard casts Maximillian's Earthen Grasp and lets the hand made from compacted soil rise from the ground on his own pillar. With another action, he wants to move the hand in order to restrain the other wizard, who is within range of 30 feet.  Would the hand retract into his own pillar, then rise again from the ground on the other pillar? Or would the hand move several kilometers down the first pillar, across the floor, then several kilometers up the second pillar? What if both pillars would float in the Astral Plane, without a direct ground-based connection between them?  This might seem like a stupid and unnecessary convoluted question, but lo and behold, something similar happened in one of my games. Two submit buttons in one form I have two submit buttons in a form. How do I determine which one was hit serverside? When rolling percentages, do 1d100 and two d10s (percentiles) share the same probabilities? Why does "kill -9 0" end my console session? I just wonder why kill -9 0 ends my tty console session?   What exactly is PID 0, it is not listed on a ps aux?