$ a , b $ are two odd integers with $\gcd(a,b)=1$. Show that $\gcd(2^a+1,2^b+1)=3$.   We have that $3\mid 2^a+1$ and $3\mid 2^b+1$ because $a$ and $b$ are odd. If we put $d=\gcd(2^a+1,2^b+1)$ I want  $3\mid d$ to conclude. Do you have any idea ?
How can I find $\gcd(a^m+1,a^n+1)$ with $a,m,n$ positive integers?    I have this idea:  Let $d=\gcd(m,n)$. Then there exist positive integers $x,y$ such that $mx-ny=d$ (WLOG). We shall find $G=\gcd(a^m+1,a^n+1)$.  If $m,n$ are odd, then $d$ is odd, therefore one and only one of $x,y$ is even. We have:  $$a^{ny}(a^d+1)=a^{mx}+a^{ny}=(a^{mx}-1)+(a^{ny}+1)=(a^{mx}+1)+(a^{ny}-1).$$  If $x$ is even and $y$ is odd, then $a^{m}+1\mid a^{mx}-1$ and $a^{n}+1\mid a^{ny}+1$, therefore $G\mid a^{ny}(a^d+1)$. If $x$ is odd and $y$ is even, then $a^{m}+1\mid a^{mx}+1$ and $a^{n}+1\mid a^{ny}-1$, thus $G\mid a^{ny}(a^d+1)$. However, since $\gcd(a^m+1,a^{ny})=\gcd(a^n+1,a^{ny})=1$, so $\gcd(G,a^{ny})=1$, hence $G\mid a^d+1$. We also have $a^d+1\mid a^{m}+1$ and $a^d+1\mid a^{n}+1$, so $a^d+1\mid G$. Thus $G=a^d+1$.  If $v_2(m)=v_2(n)=v_2(d)=k&gt;1$, then there exist some odd numbers $m_1,n_1,d_1$ such that $m=2^km_1,n=2^kn_1,d=2^kd_1$. We shall have $m_1x-n_1y=d_1$, so one and only one of $x,y$ is even, and we can use the same argument when $m,n$ are odd, so $G=a^d+1$.  However, if $v_2(m) \neq v_2(n)$, I cannot find any solutions for this. I think that $G \in \{1,2\}$, but I cannot prove or disprove it. How can I find $G=\gcd(a^m+1,a^n+1)$ if $v_2(m) \neq v_2(n)$ ? Moreover, is there anything from my arguments that can be improved ?
What is a virtual terminal for? computing the orbits for a group action Let $G$ be the Galois group of a field with nine elements over its subfield with three elements. Then the number of orbits for the action of $G$ on the fields with nine elements is   3 5 6 9   I have no idea how to compute the numbers of orbits for a group action. Anyone please help me. Thanks. I have append a mesh (rex) from this .blend .  In edit Mode I see different vertices from objcet mode (see pictures). The vertices of belt are missing!  OBJECT MODE:   EDIT MODE:   How can I edit vertices of belt? Is It a bug?  This is my .blend  maybe can help. What is the fate of the Asgardians at the start of Avengers: Infinity War? Thor: Ragnarok ends with the remaining people of Asgard aboard a refugee ship that meets Thanos' ship. Avengers: Infinity War opens with      Thanos and his minions having commandeered the ship and several bodies can be seen on the floor.   This begs the question, what happened to these characters:   Korg Miek Asguardian extra #3 Valkyrie How to concatenate multiple lines of output to one line? Best way to migrate a huge SQL Server database with low downtime over network Problem definition  Our database server needs to be transferred to an other datacenter. It runs on Microsoft SQL Server 2012 Enterprise (64-bit) and contains two databases of about 2TB and 1TB.  Having little to no downtime for this would be ideal.  Workload  Those databases are used for a .NET website and are constantly getting updated.   Having it not available over the weekend would be acceptable though. The currently in use DB would remain the only one in use until the switch over to the new one.  That switch would ideally be made just by changing DNS entries to point to the new DB server while making sure the DB is not being updated.  Also, time taken by this operation does not really matter as long as the switch from one server to the other (downtime) is kept low.  Approaches considered   Backup and restore  This has been done in the past but involved a high downtime even though it was done through an internal network, so more efficiently than through Internet Log shipping  As far as I understand, this approach would minimize downtime by configuring a master/slave and transferring  an exact copy of the master DB to its slave being read only. As mentioned above, no access to the slave would be necessary and we just need a way to have a replica of the master DB without data corruption.   It also seems to be quite efficient in terms of resources utilization and wouldn't impact to much the master performance.   I might be wrong about this approach so feel free to correct me. Database mirroring  I'm not too aware of that approach but it seems like a valid option. No need to have real time sync and performance of the master is quite important so asynchronous would be the way to go if this approach were to be chosen. Other options?  That server runs directly on bare metal hardware so lower level solutions are unfortunately not an option. Maybe there is a better way to get this done?   Constraints  As described, those databases are quite big to the point they are hard to maintain but that's an other problem.  The versions of SQL Server will be the same (Microsoft SQL Server 2012 Enterprise 64-bit).  It will have to be transferred over network between two datacenters so most probably over Internet. Having disks sent from one site to the other for an initial sync is unfortunately not an option. Having some sort of security for the transfer would be ideal but we will do the best of this situation.  That should give a quite good overview of our needs for this task and hopefully some of you had to face that situation before. There are many, many old posts on meta asking for suggestions on how to better organize favorites.  ,  and , among others.  This is not meant to be an open discussion on how to improve favorites, but rather a specific feature request that I'd like feedback on.      I would love to see a feature added that would let users arbitrarily create "groups" of favorites, and drag individual favorited questions under these groups.  I have about 2 pages of favorites (I know others have much more), and finding an individual one is quite difficult, especially since the specific question title is usually forgotten.  I just added  to my favorites, since I liked Andrew Whitaker's answer, and might want to come back and look at it later; I'd love to add this to a group called, say, "Useful Code".    I've favorited other posts since they had neat tricks in them, or because there were answers written on interesting theoretical computer science topics.  Having different groups for all these areas would be immensely helpful in finding posts later. copy ssh private keys to another computer Add the ability to add "top questions" to Careers 2.0 Profile Allow me to add my own "top questions" too! I know, I know, the site should be more heavily weighted to answering, but sometimes, how and what people ask is important too! For example I recently asked  which I think is a fairly good question. Clearly I'm going to avoid questions where I've asked something very simple, but I know of a user or two who are implementing their own C standard libraries and the depth of those questions is important/ought to be on show.  Of course, the argument I can just look at their profile holds, but that awesome q/a might have been from a year or so ago in which case finding it is difficult. How to prove a matrix is nilpotent? Given an $n\times n$ upper triangular matrix $A$ with zero on main diagonal, show that $A^n = 0$.   I did some matrix operation and noticed that the diagonal moves up, ultimately all entries will be zero. Is there a nicer way to do it? I am taking a course in Differential Topology right now, but I know of another Subject called "Riemannian Geometry" which studies Riemannian Manifolds. The definition of a real smooth manifold and a Riemannian Manifold are very similar except for the detail that makes them Riemannian (having a inner product on the tangent space of every element which varies smoothly with respect to this element of the R. Manifold).  So my question is: is there any example of a Smooth Manifold that is NOT a Riemannian Manifold?  Thanks. To better describe the above problem, whenever I start up my Ubuntu/Windows 10 Machine and select Ubuntu, whenever the machine tries to start the GNOME Desktop Environment, it says it started it successfully. however, I never get a graphical desktop environment to use. I can fix this by logging into tty3 and typing gnome-session but I don't really want to do that as, for some reason, that skips login. Can anyone help? Evaluation of $\sum_{n=0}^{\infty}\frac{1}{(n^4+n^2+1)n!}$ How do I stop mod_rewrite from appending links to url? Proofs of AM-GM inequality How to mount and unmount hard drives under Windows (the unix way) Why didn't the basilisk bite in Chamber of Secrets destroy the Horcrux? As the title asks, why didn't the basilisk bite received in the Chamber of Secrets destroy the Horcrux inside the bitten person?   Yes, I know Fawkes quite quickly healed the bite wound with his phoenix tears, but when Harry used the basilisk fang to destroy Tom Riddle's diary the destruction of that Horcrux was immediate. When Ron hit the locket with the Sword of Gryffindor, it was destroyed immediately. Although the destruction of the ring occurred off-page, Dumbledore used the Sword of Gryffindor to destroy it, so presumably, like the locket, it was gone right away. Nagini died straightaway when Neville decapitated her. The destruction of the Hufflepuff cup happens off-page. The diadem seems to have been destroyed by Fiendfyre immediately as well, as when Harry examined it outside the Room of Requirement it was already leaking a blood-like substance, and then it cracked in half. When Voldemort hit Harry with Avada Kedavra in the forest, Harry woke up in limbo with the last fragment of Voldemort's soul - that awful flayed baby-like being - separated from him immediately.   Is there a logical/canon-based explanation for why the fragment of Voldemort's soul residing in Harry wasn't destroyed upon Harry being bitten by the basilisk?  NOTE: I've read the HP Wiki's entry on Horcruxes; however, I find the HP Wiki to be inconsistent in its accuracy, so I'm hesitant to accept the Wiki's explanation on the matter. I'd prefer an explanation constructed from information in the books or a statement by JKR, or the like. Why weren't everyone on Haruun Kal made a Jedi? This is a  about securing public DNS resolvers   Open DNS servers seem pretty neat and convenient, as they provide IP addresses that we can use consistently across our company regardless of where they are located. Google and OpenDNS provide this functionality, but I'm not sure that I want these companies to have access to our DNS queries.  I want to set up something like this for use by our company, but I hear a lot about this being dangerous practice (particularly in regards to ) and I want to make sure that we do this right. What things do I need to keep in mind when building this type of environment? Set of discontinuity of monotone function is countable The result I am trying to prove is the following:     Let $F:\Bbb R\to\Bbb R$ be increasing. Then the set of points, at which it is discontinuous, is countable.   I have been reading this form Folland's Real Analysis and he proves this by considering the sum $\sum_{|x|&lt;N}[F(x^+)-F(x^-)]$ which has to be finite by some sort of telescoping sum. Now, I am not sure how he defines this sum. I am assuming he does this by using nets (let me know if there is another way to look at this), but he doesn't introduce the concept of nets until the next chapter. So I was looking for some other ways to prove this result and I found the following result (from Stein-Shakarchi) which is quite similar but assumes $F$ to be bounded.      A bounded increasing function $F$ on $[a,b]$ has at most countably many discontinuities.   This is shown by using the density of rationals in $\Bbb R$, i.e. $\exists r_x\in\Bbb Q$ s. t. $F(x^-)&lt;r_x&lt;F(x^+)$ corresponding to each point of discontinuity $x$.   Now I feel like there shouldn't be a problem in doing the exact same thing for unbounded increasing function. Is this right? Secondly how does this proof relate to the first one? I feel like there isn't much of a difference between the two proofs. To be precise I think that the previous sum is well defined/finite because of this countability of discontinuous points. What do you think about this way of thinking?  Edit. Here is the Folland's proof in more detail:  Since $F$ is increasing, the intervals $(F(x^-),F(x^+))$ are disjoint for all $\forall x\in\Bbb R$. Moreover,  $\forall|x| &lt; N$ such intervals are contained in $(F(-N),F(N^+))$ and so  $$\sum_{|x|&lt;N}[F(x^+)-F(x^-)] \leqslant F(N) - F(-N) &lt; \infty. $$ Hence the set $\{ x \in (-N,N) : F(x^+) \neq F(x^-)\}$ is countable.