I'm trying to prove the sum of reciprocals from 1 to (p-1) is divisible by p^2.  Someone showed me how to prove it. But there's a step which I don't grasp. (Or maybe I misunderstood.)  He changed the form of the sum from 1+1/2+...+1/(p-1) to p{1/(p-1)+1/2(p-2)+...+1/{(p-1)/2×(p+1)/2} and first considering only the sum of 1/(p-k)k where k is from 1 to (p-1)/2.  He said in the numerator of the sum of 1/(p-k)k , will appear every term of (p-k)k from 1 to (p-1)/2.   But I cannot show how to gurantee it.  Could you give me any hints or advice?
Write $\frac 11 + \frac12 + ...\frac1{(p-1)} =$ $\frac ab$.Such that $(a,b)=1$. Show that $p^2 \mid a$ $\text{if}$ $p\geq 5$.  I was trying to apply something with prime modulus, but I am unsure how exactly to go about it.
Java's Virtual Machine and CLR As a sort of follow up to the question called , what is the (major) differences or similarity in how the Java Virtual Machine works versus how the .NET Framework Common Language Runtime (CLR) works?  Also, is the .NET framework CLR a "virtual machine" or does it not have the attributes of a virtual machine? I had created the xml document with xml version="1.0".  In that document I need to use the greater than symbol &gt; and less than symbol &lt;.  How should I include those symbols? It's not working.  &amp;gt; and &amp;lt; are not working for me.  Is there any special encoder for this? I installed Ubuntu 17.10. Now I am having trouble with gksu:  $ gksu -dg synaptic No ask_pass set, using default! xauth: /tmp/libgksu-HgUjgQ/.Xauthority STARTUP_ID: gksu/synaptic/8760-0-alex-XPS-15-9530_TIME4974977 cmd[0]: /usr/bin/sudo cmd[1]: -H cmd[2]: -S cmd[3]: -p cmd[4]: GNOME_SUDO_PASS cmd[5]: -u cmd[6]: root cmd[7]: -- cmd[8]: synaptic buffer: -GNOME_SUDO_PASS- brute force GNOME_SUDO_PASS ended... Yeah, we're in... Unable to init server: Could not connect: Connection refused (synaptic:8767): Gtk-WARNING **: cannot open display: :1 xauth: /tmp/libgksu-HgUjgQ/.Xauthority xauth_env: (null) dir: /tmp/libgksu-HgUjgQ   If I don't use -g, the password dialog is disabled. So looks like a problem with creating a tty for root.  Any advice? How to find out why "context is incorrect"? I have been trying to write my first addon for Blender over the last couple of days. One error I came across quite often is the context is incorrect error for an operation of bpy.ops.  I learned that one can ask the poll function of an operator if the context is correct, but this does not tell why the context is correct/incorrect. What would be the best way to figure that out? Only way I could think of is to look into the code of the poll function. If that is the best way, what is the fastest way to find the appropriate piece of code? Do I need a UK visa? How and where to apply? RAM not clearing, alway in standby I'm having a problem with my ram on windows 7. The ram is clearing or programs are using and dumping it faster than it can clear. I have 8gb of ram I'm only using 4gb and the rest is in Standby. At best time I can get 100mb of free but its mostly at 15mb or less. I suspect a memory leak but I have no way to figure out what program it is. I have not installed any new hardware recently and newly installed programs have bin closed. Does any one have an idea of what is going on and how to fix it?   I'm sorry this is so little information but that is all I have as well.  Edit: I tried RamMap and found out the my "Mapped File" has 3,291,784k ram in standby how do I figure out what program and how do I limit it's ability to set ram in standby?  Edit: I keep getting a problem when I will have 0 free ram and my computer will stop working properly. It will become outrageously slow where it will take 3-4 minutes to start task manager so I can close some programmes. When I free up some ram by closing some programmes every thing runs normally. It does not matter which programs I closes only that I free up some ram. I know a lot of you say it's supposed to be like that and that's how windows works. But its obviously a problem and my computer has never worked that way neither has any of my other computers. I would like to fix this please help me. Photoshop To CSS: How do you translate Fill to CSS Evaluate $\int\frac{1}{1+x^6} \,dx$ I came across following problem     Evaluate $$\int\frac{1}{1+x^6} \,dx$$   When I asked my teacher for hint he said first evaluate     $$\int\frac{1}{1+x^4} \,dx$$   I've tried to factorize $1+x^6$ as  $$1+x^6=(x^2 + 1)(x^4 - x^2 + 1)$$ and then writing  $$I=\int\frac{1}{1+x^6} \,dx=\int\frac{1}{(x^2 + 1)(x^4 - x^2 + 1)} \,dx=\int\frac{1+x^2-x^2}{(x^2 + 1)(x^4 - x^2 + 1)} \,dx$$ $$I=\int\frac{1}{x^4 - x^2 + 1} \,dx-\int\frac{x^2}{(x^2 + 1)(x^4 - x^2 + 1)} \,dx$$  However $$x^4-x^2+1=\left(x^2-\frac12\right)^2+\frac{3}{4}$$ But I can't see how it helps  I've also tried to reverse engineer the   And I need to have terms similar to $$\frac{x^2-1}{x^4-x^2+1} \quad , \quad \frac{1}{1+x^2} \quad , \quad \frac{1}{(x+c)^2+1}\quad , \quad \frac{1}{(x+c)^2+1}$$ in integrand, How can I transform my cute looking integrand into these huge terms?  Since in exams I will neither have access to WA nor time to reverse engineer the solution moreover it does not seem intuitive,is there any way to solve this problem with some nice tricks or maybe substitutions? With constant travel, is it possible to not have a tax residence? Complete Statistic: Uniform distribution Take a random sample $X_1, X_2,\ldots X_n$ from the distribution $f(x;\theta)=1/\theta$ for $0\le x\le  \theta$.  I need to show that $Y=\max(X_1,X_2,...,X_n)$ is complete.  Now, I know I should multiply the sample distribution of $Y$ and multiply it with a function of $Y$, then integrate over the range of $\theta$ and equate them to zero. But how do I get the sampling distribution of $Y$? this principled shader node cant be shown in blender 2.78? As you can see appears "undefined" the PBT Principled shader node from blender 2.79, and I need to use it.  Is there a way or other alternative for rendering PBR Textures with all the same input settings that this shader has? I installed Ubuntu 14.04 onto my Acer Aspire E15. The PC has a 500 Gigabyte HDD.   During the installation it divided the HDD into separate partitions. However every time I turn on the PC it goes directly into Windows 8.1. It doesn't allow you the opportunity to select which OS you want to use.   Question 1: How do I set the PC so it will allow me to select the OS?  Question 2: And if #1 isn't possible. How to I remove Ubuntu from the PC and recover the partition for Windows This is a homework problem for my real analysis class. It is problem 13.1 from Mattuck's Introdution to Analysis. The question reads:       Suppose F(x) is continuous on some open interval I and c is a maximum point inside this interval. Is it true that f(x) must be increasing immediately to the left of c and decreasing immediately to the right of c? Proof or counterexample. (Note: A constant function is considered to be both increasing and decreasing.)      My friend and I have been attempting to find a counterexample. We figured a function that begins oscillating infinitely many times near c but converges to c would work. We're having difficulty constructing such a function out of elementary functions so we think we will have to define it in another way, piece-wise using receding intervals. I have no clue how this would look. Any advice would be greatly appreciated. Connecting multiple grounds As I'm relatively new to the electronics world, I was wondering if you can connect multiple  ground terminals (corresponding to several different voltage outputs) to the same ground. Or do they each need their own?  In my case I have 5 lines, one 12V, one 3.3V, two 5V and one variable from a 12V in using a  voltage regulator that already has its own ground. So I'm guessing that each would need its own ground, but it is never bad to ask. crossfade between 2 videos using ffmpeg I've been trying to achieve a crossfade transition between 2 video clips using ffmpeg but have failed so far. I'm new to ffmpeg and am mostly relying on tweaking what I can find in the documentation and existing examples online. From what I read so far, using either the blend or overlay filter should help in achieving what I'm after but I can't figure out the command line details to get it to work.  The fade and concat filters are great for fade-out of video 1, fade-in to video 2 and concat the 2 into 1 clip type transitions but I'd appreciate help in getting a command to transition from video 1 to video 2 without any going to black in between. I couldn't find any examples for exactly this problem anywhere, maybe I'm looking for the wrong keywords...?  More speficially, my videos are mp4s (h264 video, no sound, in case that matters), each is 5 seconds long and I'm after a transition from approx. 4.5s of video 1 to 0.5s of video 2.  Similar to what this tutorial does using MLT and frames (see 2:25 for an example fade), though I'm looking for a way to do this just in ffmpeg without calling any other progs.   Any pointers or maybe a command line to get a fade like this would be much appreciated, thanks very much! How do I add a non-deformed end cap to a tube? I want to create some parts that I can reuse for models. I was trying to make a steam punkish exhaust pipe. I want to be able to make the pipe of any length, and as curvy as I like but I want to keep the end cap always undeformed. I approximated my results by hand placing the "crown" at the end of this bezier curve.  I tried to make an array modifier that included the crown as the end cap, but that results in the end cap deforming. Both options affect colors in a similar way, but I can't figure out the difference. How does "failure" work in Dungeon World? How does it move the story forward? I am having trouble understanding the specifics of failure in Dungeon World. According to the rules, a GM should "make a move" against players when they fail, and that's alright, I get that.  What I don't get is how a GM is supposed to advance the plot by doing so. Let's take an example:  Rick the Fighter is climbing a massive chain that serves to hold down the wizards tower, preventing it from drifting off. He makes the move Defy Danger, to avoid slipping of, since it's raining hard, and the chain is slippery with algae and moss. He rolls and get a score of 5, meaning he fails.  In the above situation, he fails climbing the chain, so the GM should make a move. I have a hard time seeing what move would advance the story here, since I feel that the failure would imply that he doesn't get to the tower.  Is this one of those situations, where you just skip making a roll, and just let the players climb the chain, or does anyone have any idea how to handle the move, so that the story is advanced somehow?  I just want to be able to give the players some dangers that aren't traps and monsters, and I thought that a flying wizards tower was cool. I'm having a hard time seeing how I can advance the story on a failure in this situation.    Let me elaborate on the problem. I will not say that it's wrong to make the players succeed with some "cost", even on a failed roll, it just makes true "failure" impossible. That's my beef about it. The players will eventually succeed at anything that doesn't outright kill them. The Frobenius Coin Problem consider the following subsets of complex plane $$\Omega_1=\left\{c\in\Bbb C:\begin{bmatrix}1&amp;c\\\bar c&amp;1\\ \end{bmatrix}\text{ is non-negative definite } \right\} $$  $$\Omega_2= \left\{c\in\Bbb C:         \begin{bmatrix}         1 &amp; c &amp; c \\         \bar c &amp; 1 &amp; c \\         \bar c &amp; \bar c &amp; 1 \\         \end{bmatrix} \text{ is non-negative definite } \right\}$$  Let $$\bar D=\{z\in\Bbb C:|z|\le1\}$$ Then 1) $\Omega_1=\bar D,\Omega_2=\bar D$  2). $\Omega_1\neq\bar D, \Omega_2=\bar D$  3). $\Omega_1=\bar D, \Omega_2\neq\bar D$  4). $\Omega_1\neq\bar D, \Omega_2\neq\bar D$  How to proceed? Thank you.