Proving modular arithmatic congruence
For any prime $p > 3$, why is $p^2-1$ always divisible by 24?
Any data for average number of papers per year at different career stages? Does adding two linear equations will result in a line which will pass through an intersection of the linear equations? I was wondering why it is almost impossible to find a geometrical explanation of why adding two linear equations helps us to find a solution of a system of linear equations? Am I right that adding two linear equations will result in an equation of a line which will pass through a point where two linear equations intersect?  If it is right then I completely don't understand why such a crucial point in understanding how to solve systems of equations is never properly taught even at a university level. Is it OK for OSB3 boards to get wet during construction? proving $\left(1+\frac 1n\right)^{n} = 1 + \sum_{k=1}^n \{\frac 1{k!}\prod_{r=0}^{k-1}(1-\frac rn)\}$ using the binomial theorem Let $a,b,c,x,y,z&gt;0$ and $x+ y+ z= 1$. Show that $$ax+ by+ cz\geq  a^{x}.b^{y}.c^{z}$$ I have found the generalization of this inequality: For 2 sets of numbers $\left \{ a_{i} \right \}, \left \{ b_{i} \right \}&gt; 0$ and $\sum_{1}^{n}b_{i}= 1$, we have: $$\sum_{1}^{n}\left ( a_{i}b_{i} \right )\geq  \prod_{1}^{n}\left ( a_{i}^{b_{i}} \right )$$ First solution Simply compare the logs: as the natural logarithm is a concave function, one has $$\ln\bigl(a^x b^y c^z)=x\ln a+y\ln b+z\ln c\le \ln(ax+by+cz)$$ for all non-negative $x,y,z$ such that $x+y+z=1$. How about another solution? I really hope to see that. Thanks! I'm in trouble with time dilation: Suppose that there's two people on the Earth (A,B), they are twins and each other has a clock. (So they are at the same reference frame). B travels in a spaceship and is orbiting around the Earth, as B's speed has increased there's some time dilation. So it's supposed that if A "looks" to B, B's clock will run slower than A's, and viceversa. And if one day B decides come back to the Earth, he will be younger than his twin and the hour of his clock would be different (earlier than A's clock) because of time dilation.  My first question is, why each person see the other's clock as the slow one? (In fact, I found some information that may solved it but I'm not sure if it's the right answer, and if it is, I don't understand it. -> It's because everyone thinks about himself as the reference frame. In other words, that for B, he is the stationary and who moves is A, so he sees A's clock run slower as B is "stopped" and the same for A. B is who is moving and A the stationary.)  The other question I have is that if it's true that each one sees the other as the slow one there's something that i missed out. We all agree that B (after going to the space and came back to the Earth) he is younger than his brother. So, why B don't see A moving and aging very fast if A see his brother and his things going very slow? Let's imagine they were looking to each other all the time, if both of them see the other going slowly once they meet and "discover" B is the young one... how is that possible?  [I think that in one episode of Cosmos of Carl Sagan (though may be I'm mistaken) he said that a neutrino "borned" in the Big Bang could have seen the creation of the Universe until today in few seconds due to its high speed, here starts my doubts: or i misunderstood something or there's contradictory information]  (I don't know anything about physics, only what is taught at high school as I'm a teenager so it'd be better to answer with no kind of calculation as I'd not understand it.) Question I use pgfplots with the ultra thick option ans without explicitly giving the domain for the y-axis. It seems that in this case, some part of my graph is &quot;eaten&quot; by the margin: see the picture below to understand what I mean. Is there a way to avoid this? Code \documentclass{article} \usepackage{pgfplots} \begin{document}  \begin{tikzpicture}  \pgfplotsset{every axis/.append style={                     axis x line=middle,    % put the x axis in the middle                     axis y line=middle,    % put the y axis in the middle                     axis line style={-&gt;},  % arrows on the axis                     xlabel={$x$},          % default put x on x-axis                     ylabel={$y$},          % default put y on y-axis                     grid = major                     }}  \begin{axis}[xmin=-5,xmax=5] \addplot[orange,smooth, ultra thick] {sin(deg(x))} ;                  \end{axis} \end{tikzpicture}  \end{document} Dividing an equilateral triangle into N equal (possibly non-connected) parts It’s easy to divide an equilateral triangle into $n^2$, $2n^2$, $3n^2$ or $6n^2$ equal triangles. But can you divide an equilateral triangle into 5 congruent parts? Recently M. Patrakeev  an awesome way to do it — see the picture below (note that the parts are non-connected — but indeed are congruent, not merely having the same area). So an equilateral triangle can also be divided into $5n^2$ and $10n^2$ congruent parts.  Question. Are there any other ways to divide an equilateral triangle into congruent parts? (For example, can it be divided into 7 congruent parts?) Or in the opposite direction: can you prove that an equilateral triangle can’t be divided into $N$ congruent parts for some $N$?                                               (Naturally, I’ve tried to find something in the spirit of the example above for some time — but to no avail. Maybe someone can find an example using computer search?..) I’d prefer to use finite unions of polygons as ‘parts’ and different parts are allowed to have common boundary points. But if you have an example with more general ‘parts’ — that also would be interesting. How is exponentiation defined in Peano arithmetic? How would exponentiation be defined in Peano arithmetic? Unless $n$ is fixed natural number, $x^n$ seems to be hard to define.   Edit 2: So, what would be the way to define $x^n+y^n = z^n$ using $\Sigma_1^0$ formula?  Edit: OK, I say Peano arithmetic has addition and multiplication stuffs. This allows additions to be expressed without quantifiers but as for exponentiation Peano arithmetic is silent. That's why I asked this question. Just for clarification. Why does the square root of a square involve the plus-minus sign? If $\sqrt{x^2}$ can be simplified as follows:  $\sqrt{x^2} = (x^2)^\frac{1}{2} = x^{\frac{2}{1}\times\frac{1}{2}} =x^\frac{2}{2} = x^1 = x$  Then why would $\sqrt{x^2} = \pm x$? Wacom Tablet - Illustrator brush tool I currently got the intuos art wacom tablet so still learning how to use it. I'm currently in illustrator trying to draw out a map and the brush tool is doing something weird and I cannot figure out why. You can see in my image that where there are curves the black slightly spills over and is filling in half the curve with black. Brush setting is set up as a new calligraphic brush and the size is set to pressure, 3pt-3pt. Are the units of the quotient ring $\mathbb{F}_2[x]/\langle x^k+1 \rangle$ known in general, where $\mathbb{F}_2$ is the finite field with two elements? I'm specifically interested in the case where $k$ is divisible by two, such as say $k=8$, or $k=12$.  If so, can they be constructed easily? I'd be grateful if you could provide a few examples. Getting polygon boundaries of City in JSON from Google Maps API? When searching Google Maps for a city, the city's administrative boundary is displayed.    Is there a way to get the polygon boundaries of City in a JSON format using google maps api? How do I get started playing Dungeons and Dragons 4th edition by myself? I've been wanting to start playing D&amp;D for some time, and I was wondering how I could get started by myself with relative ease. I own the first two player handbooks, and a set of dice. Rendering an animation the way 3D view looks Is there some way to render exactly whats in the camera view in the 3D view? By that I mean just like a screenshot really, all paths, lamps, and cameras, just like things look in the camera view. The reason I can't just take a screenshot is that I need an animation and it would be too cumbersome to take 1000+ screenshots! How to replace ${} placeholders in a text file? Why is $2\pi i \neq 0?$ We know that $e^{\pi i} = -1$ because of de Moivre's formula. ($e^{\pi i} = \cos \pi + i\sin \pi = -1).$  Suppose we square both sides and get $e^{2\pi i} = 1$(which you also get from de Moivre's formula), then shouldn't $2\pi i=0$? What am I missing here? I'm running Ubuntu server on a computer used as a wireless AP, but this AP should resolve all DNS requests to an internal IP address rather than actually performing the lookup.  I want to do the same thing that paid public WiFi hotspots do - you can connect but if you attempt to load any websites they show a default page. I've noticed that they do this by resolving all domains to an internal IP address.  I've added these lines to /etc/dnsmasq.conf:  # Add domains which you want to force to an IP address here. # The example below send any host in double-click.net to a local # web-server. address=/com/192.168.2.1 address=/uk/192.168.2.1 address=/org/192.168.2.1 address=/gov/192.168.2.1 address=/net/192.168.2.1 address=/us/192.168.2.1   which works fine for those TLD's, but I'd like to be able to do it with all domains so I can sleep at night. Which race in Star Trek universe was the first to invent Warp technology? Humans look like late member of Warp Drive community. Klingons were capable of interstellar travel from the time of Kahless. Vulcans were in contact with Andorians from ancient times (ST: Enterprise).  Which one first invented warp capabilities: Klingons or Vulcans? Or, others? Can "who" as a relative pronoun sometimes be omitted?