How can I prove that m, n in $m^2 = 6n^2$ are not integers?
Verify my proof of the irrationality of $\sqrt{6}$.
Can PS4 Minecraft join with PC Minecraft? I have been thinking of getting a flash to improve my low light photography (particularly indoors). However, there seem to be a number of different models available at similar price points (budget of INR 8,000-12,000 ~$150-250). What features and functionality should I look for to choose a suitable flash?  I'm using a Canon 550D with the 18-55mm IS &amp; 55-250mm IS lenses. Metz seems to be one of the brands easily available in India (beside the OEM flashes). Currently, I am considering the Metz 36 AF-5, 44 AF-1 and 50 AF-1, which seem to be considerably cheaper than their Canon counterparts. Python module for converting PDF to text Is there any python module to convert PDF files into text? I tried  found in Activestate which uses pypdf but the text generated had no space between and was of no use. Let $R$ be a commutative ring and $I_1, \dots, I_n$ pairwise comaximal ideals in $R$, i.e., $I_i + I_j = R$ for $i \neq j$. Why are the ideals $I_1^{n_1}, ... , I_r^{n_r}$ (for any $n_1,...,n_r \in\mathbb N$) also comaximal? When I receive a message on an SE post, I get a notification like this:    If I double tap this notification, the SE application opens, but my feed is empty approximately half the time. It looks like this:    If I refresh the feed (slide from the top), the feed does refresh and shows the new notification.   The notification, above, leads to  comment.  I haven't figured out the pattern yet, but this occurs roughly half the time I open a notification. It's occurred for chat messages, comments and answers - but not consistently. I have received notifications for the same post (for example, the two comments after the one linked above) that do not show an empty feed. But, a chat message barely 10 minutes later will show the empty feed again.  I am utilizing the beta app - version 1.0.72 on Android 5.0.1. Prove that $T$ has an orbit of period 3 If you draw two cards, what is the probability that the second card is a queen? We had this question arise in class today and I still don't understand the answer given. We were to assume that drawing cards are independent events. We were asked what the probability that the second card drawn is a queen if we take two from the deck. The answer given was 4/52, which seems counter-intuitive to me. How is the probability still 4/52 if there was a card drawn before it? What if the first card drawn was a queen? If you get stuck in research, at what point in time should you ask for help? It's always worthwhile to take some time to try to figure it out by yourself. But eventually there's a point where you're unsure how much more time it would take to figure it out by yourself (which could take far longer than you originally think), and when asking for help could be more helpful.  So for analytical or computational research, what are some guidelines for when you should ask for help (from either your adviser, other grad students, or other researchers) when you get stuck? What are irreducibles in the ring $R$ of entire functions? Here's how far I have gone:  I first find the units in $R$. Let $f$ $ \in $ $R$ be a unit. Then $\exists$ $g$ $\in$ $R$ such that $fg=1$, i.e $f(z)g(z)=1$ $\forall$ $z$ $\in$ $\Bbb C$.  Then, $f(z)$ $=$ $\frac 1{g(z)}$ $\forall$ $z \in \Bbb C$. Hence $f(z),g(z) \neq 0$ $\forall$  $z \in \Bbb C$.  Now to find the irreducibles, let $f \in R$ be an irreducible. Hence whenever  $f(z)=g(z)h(z)$ for $g,h \in R$, then $g$ or $h$ is a unit.  Let $h$ be the unit and $\alpha$ be a zero of $f$. Then $f(\alpha)=g(\alpha)h(\alpha)=0$ $\Rightarrow$ $g(\alpha)=0$. Hence $z-\alpha$ is a factor of $g$. So $g(z)=(z-\alpha)^ng_1(z)$ for some $n \in \Bbb N$, where $g_1$ is unit. Since $f$ is irreducible, hence $n=1$.  I'm stuck here. What can be my next step? Why can't variables be declared in a switch statement? I know that : $\mathbf{Q}(\sqrt{2}) = \mathbf{Q}+ \sqrt{2} \mathbf{Q}$ , but then what is $\mathbf{Q}(\sqrt{2},\sqrt{3})$? Is "everyone" singular or plural? Which is correct?     Everyone were convinced that he would go to the game.   Everyone was convinced that he would go to the game.   I think it's "was", because "everyone" is singular, but I just wanted to check. Find an equation in spherical coordinates for the surface represented by the rectangular equation The rectangular equation is $$x^2+y^2-8z^2=0$$ $$x^2+y^2=8z^2$$ Know in the relationship between rectangular and spherical coords. we can manipulate our given to fit the form: $$x^2+y^2+z^2=9z^2$$ $$\rho=x^2+y^2+z^2, \space z=\rho\cos(\phi)$$ $$\rho^2=9\rho^2\cos^2(\phi)$$ $$1=9\cos(\phi)$$ $$\frac{1}{3}=\cos(\phi)$$ $$\arccos(\frac{1}{3})=1.23 \space rads$$ And so the equation in spherical coords. is $\phi=1.23$ I know my math is correct but I have the wrong answer so I'm not sure where I went wrong. Resampling GeoTIFF images to same resolution using QGIS I have two geotiff images that I would like to resample the same resolution. If I use ArcGIS Desktop this is easy to do but how do I do this using QGIS? Trigger Task Scheduler in Windows 7 when computer wakes up from sleep/hibernate I don't want task scheduler or whatever to wake the computer, but when I open the laptop back up and the machine wakes I'd like that to be the trigger that task scheduler or something uses to run an app. Side effects of changing blackboard bold font with \setmathfont I have a register \count0=&lt;some-value&gt; and need to get the letter of the alphabet on the index of \count0. That is, if it is 1, I need an A, if 2, a B, etc.  Is this possible?  I'm reluctant sharing an MWE as it seems to overcomplicate things a bit:  \begin{tikzpicture} \node[regular polygon, regular polygon sides=7, minimum size=5cm] (A) {}; \foreach \i in {1,...,7}{     \count0=8     \advance\count0 by -\i     \node[circle, label=above:\scriptsize\the\count0, fill=black, minimum size=4pt, inner sep=0pt] (\the\count0) at (A.corner \i) {}; } \end{tikzpicture}   With the tikz package and the shapes.geometric tikz library.  This draws seven nodes labelled 1 .. 7 on a circle. I need to have A .. G instead of 1 .. 7. I often hear about the wave-particle duality, and how particles exhibit properties of both particles and waves. However, I wonder, is this actually a duality? At the most fundamental level, we 'know' that everything is made up out of particles, whether those are photons, electrons, or maybe even strings. That light for example, also shows wave-like properties, why does that even matter? Don't we know that everything is made up of particles? In other words, wasn't Young wrong and Newton right, instead of them both being right? If I know that a certain key has been pressed (eg Key.D3), and that the Shift key is also down (Keyboard.IsKeyDown(Key.LeftShift) || Keyboard.IsKeyDown(Key.RightShift)), how can I find out what character that refers to (eg, # on US keyboard, UK pound sign on UK keyboard, etc)?  Put another way, how can I find out, programatically, that Shift + 3 produces # (it wouldn't on a non-US keyboard). How to force restart an iPhone with broken home button Earlier today I was on my iPhone and it randomly froze so I turned it off. When I went to unlock it it wouldn't move. I couldn't unlock it or tap anything. So now I can only stay on the lock screen. I read on the Apple website that I should "force restart" it by pressing my home button and power button at the same time. The problem is my home button is stuck and doesn't work. What should I do?