Establish that 7 is a primitive root of any prime of the form $p = 2^{4n} + 1$.
For every prime of the form $2^{4n}+1$, 7 is a primitive root.
How to filter uninteresting questions Convergence of $\sum_{n=1}^\infty\frac{n}{(n+1)!}$ Can someone give an explanation using the definition of convergence in partial sum to show how the above infinite sum converges to 1?  Thanks Opposite of "straight talk" This morning, I had eggs for breakfast, and I was looking at the pieces of broken shells and thought "What is the surface area of this egg?" The problem is that I have no real idea about how to find the surface area.    I have learned formulas for circles, and I know the equation for an ellipse; however, I don't know how to apply that.   The only idea I can think of is to put an egg on a sheet of paper and trace it, and then measure the outline drawn, and then try to find an equation for that ellipse and rotate that about the $x$-axis.  Now, my problem is how I can find the equation of the ellipse from the graph, and will my tracing method really be the edge of the egg?  Also, can I use the ? Will I have to use some techniques to solve the integral that are not covered in the ?  There has to be a better method for finding the surface area. Please, help me understand how to find the surface area of an egg; i.e., how to use my mathematical knowledge for something other than passing exams. I can not see how mathematically one can divide the numerator and denominator by theta. Can some one explain this in a different way. When you divide two fractions you multiply the first fraction by the reciprocal of the second. This is not the case here.     Divide numerator and denominator by $\theta$. ($\sin\theta$ also works.) $$\eqalign{\lim_{\theta\to0}\dfrac{\sin\theta}{8\theta+\tan\theta}&amp;= \lim_{\theta\to0}\dfrac{\dfrac{\sin\theta}\theta}{8+\dfrac{\sin\theta}\theta\cdot\dfrac1{\cos\theta}}\\&amp;=\dfrac{\lim\limits_{\theta\to0}\dfrac{\sin\theta}\theta}{8+\lim\limits_{\theta\to0}\dfrac{\sin\theta}\theta\lim\limits_{\theta\to0}\dfrac1{\cos\theta}}\\&amp;=\dfrac1{8+1\cdot1}\\&amp;=\dfrac19.}$$ Why should I "watch" a nearby Pokémon? Kubuntu - can not enter wifi credentials after adding external wifi card My ethernet connection is quite unreliable (It uses powerline adapters) so I bought an external wifi adapter. I have plugged it in and run the command sudo ip address add 192.168.0.39/24 brd + dev enp7s0, and KDE has detected it. Unfortunately, it doesn't seem to realise it is a wireless network and therefore needs credentials. Where I could normally find a place to enter credentials (the network widget, system settings) there are none. KDE reports that the device is connected in the UI. How do I make it work out that it is wireless and therefore needs credentials? Edit: With ethernet plugged out, (USB) adapter plugged in: $ ip address show 1: lo: &lt;LOOPBACK,UP,LOWER_UP&gt; mtu 65536 qdisc noqueue state UNKNOWN group default qlen 1000     link/loopback 00:00:00:00:00:00 brd 00:00:00:00:00:00     inet 127.0.0.1/8 scope host lo        valid_lft forever preferred_lft forever     inet6 ::1/128 scope host         valid_lft forever preferred_lft forever 2: enp7s0: &lt;NO-CARRIER,BROADCAST,MULTICAST,UP&gt; mtu 1500 qdisc fq_codel state DOWN group default qlen 1000     link/ether 04:d9:f5:8a:f5:e0 brd ff:ff:ff:ff:ff:ff  With ethernet plugged in, adapter plugged in: 1: lo: &lt;LOOPBACK,UP,LOWER_UP&gt; mtu 65536 qdisc noqueue state UNKNOWN group default qlen 1000     link/loopback 00:00:00:00:00:00 brd 00:00:00:00:00:00     inet 127.0.0.1/8 scope host lo        valid_lft forever preferred_lft forever     inet6 ::1/128 scope host         valid_lft forever preferred_lft forever 2: enp7s0: &lt;BROADCAST,MULTICAST,UP,LOWER_UP&gt; mtu 1500 qdisc fq_codel state UP group default qlen 1000     link/ether 04:d9:f5:8a:f5:e0 brd ff:ff:ff:ff:ff:ff     inet 192.168.0.51/24 brd 192.168.0.255 scope global dynamic noprefixroute enp7s0        valid_lft 863988sec preferred_lft 863988sec     inet6 fe80::7eb1:eca2:96a0:fe5c/64 scope link noprefixroute         valid_lft forever preferred_lft forever  With just ethernet: 1: lo: &lt;LOOPBACK,UP,LOWER_UP&gt; mtu 65536 qdisc noqueue state UNKNOWN group default qlen 1000     link/loopback 00:00:00:00:00:00 brd 00:00:00:00:00:00     inet 127.0.0.1/8 scope host lo        valid_lft forever preferred_lft forever     inet6 ::1/128 scope host         valid_lft forever preferred_lft forever 2: enp7s0: &lt;BROADCAST,MULTICAST,UP,LOWER_UP&gt; mtu 1500 qdisc fq_codel state UP group default qlen 1000     link/ether 04:d9:f5:8a:f5:e0 brd ff:ff:ff:ff:ff:ff     inet 192.168.0.51/24 brd 192.168.0.255 scope global dynamic noprefixroute enp7s0        valid_lft 863960sec preferred_lft 863960sec     inet6 fe80::7eb1:eca2:96a0:fe5c/64 scope link noprefixroute         valid_lft forever preferred_lft forever  With nothing: 1: lo: &lt;LOOPBACK,UP,LOWER_UP&gt; mtu 65536 qdisc noqueue state UNKNOWN group default qlen 1000     link/loopback 00:00:00:00:00:00 brd 00:00:00:00:00:00     inet 127.0.0.1/8 scope host lo        valid_lft forever preferred_lft forever     inet6 ::1/128 scope host         valid_lft forever preferred_lft forever 2: enp7s0: &lt;NO-CARRIER,BROADCAST,MULTICAST,UP&gt; mtu 1500 qdisc fq_codel state DOWN group default qlen 1000     link/ether 04:d9:f5:8a:f5:e0 brd ff:ff:ff:ff:ff:ff  lsusb, adapter plugged in: Bus 004 Device 001: ID 1d6b:0003 Linux Foundation 3.0 root hub Bus 003 Device 002: ID 046d:c31c Logitech, Inc. Keyboard K120 Bus 003 Device 001: ID 1d6b:0002 Linux Foundation 2.0 root hub Bus 002 Device 001: ID 1d6b:0003 Linux Foundation 3.0 root hub Bus 001 Device 003: ID 1d57:32da Xenta 2.4GHz Receiver (Keyboard and Mouse) Bus 001 Device 004: ID 0bda:b812 Realtek Semiconductor Corp. 802.11ac NIC Bus 001 Device 001: ID 1d6b:0002 Linux Foundation 2.0 root hub  I'm not adding it with it plugged out because it's obviously the Bus 001 Device 004: ID 0bda:b812 Realtek Semiconductor Corp. 802.11ac NIC. There were no restarts between any of these changes. Retrieving Maxwell's equations from the minimum action principle I'm currently working at the start of Alexei Tsvelik's book Quantum Field Theory in Condensed Matter Physics. I'm kinda stumped on a few essential steps.  Starting with the action:  $$S = \int dt \int d^3 x \mathcal{L}_{\mathrm{EM}} = \int dt \int d^3 x \left( -\frac{1}{4} F_{\mu \nu} F^{\mu \nu} + j_\mu A^{\mu}\right).$$  In the absence of any sources $j_\mu = 0$ and $c=1$, which after pulling in the negative sign implies  $$S = \frac{1}{2}\int dt \int d^3 x  \,\,\left(E^2 - B^2\right)$$ The next step is to take $\vec{E} = -\vec{\nabla} \phi + \partial_t \vec{A}$ and $\vec{B} = \vec{\nabla} \times \vec{A}$. Letting $\phi = 0$ and directly inserting these into the above yields:  $$S = \frac{1}{2} \int dt \int d^3 x  \,\left( (\partial_t \vec{A})^2 - (\vec{\nabla} \times \vec{A})^2 \right) $$  Now considering $\vec{A} = \vec{A}_0 + \delta \vec{A}$ as some minimal change in our vector potential, with $\vec{A}_0$ gives a minimum for the action and putting this into the above expression we have: $$\delta S = \int dt \int d^3 x \left(\partial_0 \vec{A}_0 \partial_0 \delta \vec{A} + \left(\vec{\nabla} \times \vec{A}_0\right)\cdot\left(\vec{\nabla} \times \delta \vec{A} \right) \right)+ \mathcal{O}(A^2)\,\,\,\,\,\,\, (1)$$ where I have the term  $$\mathcal{O}(A^2) = \int dt \int d^3 x \left( (\partial_0 \vec{A}_0)^2 + (\partial_0 \delta \vec{A})^2 - (\vec{\nabla}\times\vec{A}_0)^2-(\vec{\nabla}\times \delta\vec{A})^2\right)$$  The next step is where I lose our author and go from "Oh ok I get this" to "completely clueless". He rewrites $\delta S$ as:  $$\delta S =  \int dt \int d^3 (\delta x) \vec{A} \mathcal{F(\vec{A}_0)}+ \mathcal{O}(A^2)$$  where $\mathcal{F} = \frac{\delta S}{\delta A}$. What is the purpose of $x\to \delta x$ in the above expression? Is the functional derivative $\mathcal{F}$ a vector quantity? He has it bolded with a bolded $\vec{A}$ in the above expression. One last question takes the above expression further. We assume $\delta \vec{A}$ vanishes at infinity, as it should, and integration by parts yields:  $$\delta S =  -\int dt \int d^3 x \left( \partial^2_0 \vec{A}_0 - (\vec{\nabla} \times \vec{\nabla})\times \vec{A}_0  \right)\delta \vec{A}\,\,\,\,\,\,\,\, (2)$$ How exactly did we go from (1) to (2)? Thanks I appreciate your time! Number of planks required to stop the bullet I wandered too far to go look for sheep and now I'm on a completely different island.  How do I find my way back to my house? Not getting Rep Notification for new up votes So just a little while ago i was flipping out because i was 2 rep away from my 10k rep goal    but then all of a sudden it was then 10,008 rep because someone upvoted me. however i didn't get any notification about this +10  [    now as you can see in the screenshots above, over the past 2 days i got 2 upvotes to a question i posted, however the notifications only shows 1 +10 notification. and they system knows i got the upvote because i knew that i now have access to mod tools.  I am on a Windows 10 Laptop using Chrome 78.0.3904.97 (Official Build) (64-bit) ISP blocking port 25 Prove $\gcd(a,b) \gcd(a,c) \gcd(b,c) \,\text{lcm} (a,b,c)^2=$ $\text{lcm}(a,b)\,\text{lcm}(a,c) \,\text{lcm}(b,c) \gcd(a,b,c)^2$ I'm trying prove $$\gcd(a,b) \gcd(a,c) \gcd(b,c) \,\text{lcm} (a,b,c)^2=\text{lcm}(a,b)\,\text{lcm}(a,c) \,\text{lcm}(b,c) \gcd(a,b,c)^2$$  It is a part of solution of another problem I try a lot to prove it but I could not be succeed.  I tried using exponent manipulation in prime factorizations but it didn't help me. Vueling is currently experiencing , with many flights delayed by several hours. Assuming I'm at the airport and my flight is certainly delayed by more than 3 hours, can I simply book another flight and claim compensation later?  According to EU's :     In addition, if you are denied boarding, your flight is cancelled or arrives more than 3 hours late on arrival at the final destination stated on your ticket, you may be entitled to compensation of EUR 250 - 600, depending on the distance of the flight.   But it's not clear if I actually have to board the flight or if simply having a purchased ticket is sufficient. I presume it would be possible to negotiate with Vueling representatives at the airport, but I'd rather not wait in the inevitable 100-person line to their counter if I'm covered by EU law. I have a local user on my laptop. My company is migrating to a domain setup and I want to migrate the local user to a domain user, with all the settings (desktop, Start menu, programs) intact. How can I do that?  Also, is it possible to have a local user with the same settings as the domain, for when I'm working on my laptop while not connected to the domain? How to make the taskbar blink my application like Messenger does when a new message arrive? "Not a COM32R image" error when trying to install from a USB key Not a COM32R image boot   is a message that the console infinitely keeps spitting out when I try to install Ubuntu 10.04 from a USB. I created the USB using the StartupDiskCreator in 11.04.  What is the reason for this behavior? What do I do to set it right?   I have followed the steps perfectly, as enlisted in the . The  badge seems like a great candidate for a trackable badge: it requires doing a thing multiple times, there aren't any other requirements to get it, and it is (seems?) quite easy to determine how many times the thing has been done.  But Convention does not appear in the list of badges that can be tracked. I would like to be able to see a quick count of my well-received meta posts without going to my meta user activity page and poking around. Does "no larger than" imply shapability? I remember reading a while back about arithmetics with only a single operator that did all the work of the familiar $+ - \times /$ operators. If there is such a thing, could someone point me to it?  As background: I have a side project where I'm playing with neural nets and genetic search algorithms. Right now it's the good old kind, with matrix multiplication and addition. I thought it would be interesting to implement a sort of neural network that used a combined or "all in one" operator.  I've googled like heck and can't pick up the trail of where I thought I saw that before.  Is there study in that area, or am I mis-remembering, or worse, just making it up? I'd appreciate any pointers.