Number of positive integral solutions $(x,y)$ of $x^2-y^2=12345678$
Does $x^2 - y^2 = 12345678$ (x,y are integers) have any solutions?
Let $f(z)$ be an entire function such that $$|f(z)|&lt;\frac{1}{|\text{Im}(z)|},\qquad z\in\Bbb C-\Bbb R.$$ The question asked me to prove that $f(z)=0$. At least looking at it, it really seems to have an application of Liouville's theorem lurking around somewhere, but I haven't found it.   My thoughts first led me to think about doing this by contradiction and using Picard's little theorem. So, I've considered a strip containing the real axis (say of width $2$ for simplicity). This will imply the complement maps into the unit disk, which implies that this strip has to map almost everywhere else on the complex plane. Now, on the imaginary axis, I know that $f(z)$ will vanish as $z\to\infty$, and this seems like it might be useful. I'm sort of under the impression $f(z)$ might have an essential singularity at $\infty$, which would mean $f(1/z)$ has an essential singularity at $0$. Either way, it of course can't have a pole at $\infty$ because of $f(z)$ vanishing on the imaginary axis, and if it is a removable singularity, it must be $0$, which still gives the solution. Therefore, I think it must have something to do with $f(z)$ having an essential singularity at $\infty$.   I was hoping to combine this with the above inequality and deduce a contradiction, but I haven't thought of one yet.   Any hints or suggestions? I'm studying for a prelim. exam, so this isn't homework.  Update: So, in the spirit of searching methods using the essential singularity at $z=0$, we have $$\left|f\left(\frac{1}{z}\right)\right|\leq\frac{1}{\left|\text{Im}\left(\frac{1}{z}\right)\right|}=\frac{|z|^2}{|y|}.$$ Thus, for $|y|&gt;1$, we have $$\left|f\left(\frac{1}{z}\right)\right|&lt;|y|\left|f\left(\frac{1}{z}\right)\right|&lt;|z|^2.$$ Thus, if I can show that $|f(1/z)|$ is a polynomial, we'll know it can't have an essential singularity at the origin, concluding my proof. Is my paper under review (or similar) for too long and if yes, how should I react? What's the intuition behind difference of $\neg(S\land E)$ and $\neg S \land \neg E$? Let $S$ represent the English statement "Sales will go up."And let $E$ represent the English statement "Expenses will go up." Now my problem is that I have trouble telling the difference between these two statements: $\neg(S\land Q)$ and $\neg S \land \neg E$. When these two statements are translated into English they sound,to me, the same. However, when you look at their truth tables they are different.   Here is my best translation of these two statements:  1) $\neg(S\land E)$ represented in English, "Both sales and expenses will not go up."(Correct me if I translated it wrong)  2)$\neg S \land \neg E$ represented in English, "Expenses will not go up and sales will not go up."(Again, correct me if I'm wrong)  I still do not understand the difference between these two statements in the English representation. They both sound the same thing to me.  Once more, I have the same trouble I have with understanding with the or-connective( $\lor$ ). Again let E and S be the same statements as before.Tell me if I might have translated them wrong.  1) $\neg (S \lor E)$ represented in English is "Neither sales nor expenses will go up."  2)$\neg S \lor \neg E$ represented in English is "Sales will not go up, or Expenses will not go up."  Yet I still have trouble understanding the difference between the English translation.  In short, can you give me some alternative example that better explains the difference, something intuitive that gives me a clear picture of the difference of the meaning between the statements I showed with the connective-or and connective-and. Thank you. There are Recurrent Neural Networks and Recursive Neural Networks. Both are usually denoted by the same acronym: RNN. According to , Recurrent NN are in fact Recursive NN, but I don't really understand the explanation.  Moreover, I don't seem to find which is better (with examples or so) for Natural Language Processing. The fact is that, although Socher uses Recursive NN for NLP in his , I can't find a good implementation of recursive neural networks, and when I search in Google, most of the answers are about Recurrent NN.  Besides that, is there another DNN which applies better for NLP, or it depends on the NLP task? Deep Belief Nets or Stacked Autoencoders? (I don't seem to find any particular util for ConvNets in NLP, and most of the implementations are with machine vision in mind).  Finally, I would really prefer DNN implementations for C++ (better yet if it has GPU support) or Scala (better if it has Spark support) rather than Python or Matlab/Octave.  I've tried Deeplearning4j, but it's under constant development and the documentation is a little outdated and I can't seem to make it work. Too bad because it has the "black box" like way of doing things, very much like scikit-learn or Weka, which is what I really want. What is bracketing? I have seen this term a lot and I was wondering what, in basic terms, what bracketing/autobracketing is.  Also, why would one want to 'bracket'?  P.S. - there are other questions asking what DoF and WB bracketing is but none asking quite the same as this. What is "umask" and how does it work? I believe that umask is something that controls , but do not fully understand it.  After running umask 0644 in a , I cannot read the files I create with the command-line text editor nano. I noticed that the permissions of that file are set to 0022 instead of the default 0755.  How does umask work? I thought I could just remove the each digit in the umask from 0777, 7 - 6 = 1 and 7 - 4 = 3, so I expect the permissions to be 0133, but apparently, this is not the case.   What is umask exactly? Explain it to me like I was a "Linux noob" How do I calculate with umask? What are use cases for umask? Can you concentrate on a special ability while also concentrating on a spell? There exist several abilities that have the text similar to, this effect lasts while you maintain concentration (as if you were concentrating on a spell). One such example is the Bard College of Glamour: Mantle of Majesty ability.     At 6th level, you gain the ability to cloak yourself in a fey magic   that makes others want to serve you. As a bonus action, you cast   command, without expending a spell slot, and you take on an appearance   of unearthly beauty for 1 minute or until your concentration ends (as   if you were concentrating on a spell). During this time, you can cast   command as a bonus action on each of your turns, without expending a   spell slot.   Another example is the Cleric Trickery Domain: Channel Divinity: Invoke Duplicity ability.     Starting at 2nd level, you can use your Channel Divinity to create an   illusory duplicate of yourself.      As an action, you create a perfect illusion of yourself that lasts for   1 minute, or until you lose your concentration (as if you were   concentrating on a spell).   While you are concentrating on maintaining such an ability would you also be able to maintain concentration on a spell that requires concentration?  There does not seem to be any RAW that limits this however the limitation around concentrating on a second spell exists which may suggest that the intent is that a character cannot concentrate on 2 effects at the same time unless explicitly stated. Replace a long string with the sed command: Argument list too long error I'm trying to execute this command  sed -i -e "s/BASE_64/$BASE_64/" FILE_NAME   where $BASE_64 is a base 64 representation of a file content.  sed gives me an error since the string is too long.     Argument list too long   How is it possible to avoid this error? Rocket Altitude Calculation Assignment Basic data I was launching a rocket model and I tried to calculate the reached altitude.  The engine (C6-0) impulse is 10 Ns Total weight is 65,7 g (includes the engine)  I calculated speed = 152 m/s $$\vec F \cdot t = \Delta m \vec v$$ Then I calculated the altitude 1007 m which seems too much to me. I guess something about 200 m (you may see the video) $$y_{\max} = \frac{v_0^2 \ }{2 g}$$ Drag I guess, I have to consider drag  Diameter of rocket 2,5 cm Drag coefficient 0,05 (I guess)  $$F_D\, =\, \tfrac12\, \rho\, v^2\, C_D\, A$$ But what about now, what is really achieved altitude? Engine description  (source: )  (source: ) Rocket Altitude Calculation Unchained Flurry of Blows with Natural Weapons and Feral Combat Training using natural attacks as primary Make HttpURLConnection load web pages with images Display command in xterm titlebar My Bash Prompt is currently setting the xterm titlebar using the following sequence:  PS1='\033]0;\u@\h&gt;\w\007'   Is there an easy way to display the current command in the titlebar. For example, if I am tailing a file using tail -f foo.log, I want my titlebar to say tail -f foo.log. Why is $\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}$? Flames with no gravity? I was watching  today, and noticed this: in some moment the space station changed orbit and the people inside experienced zero-gravity. At that moment, a candlestick passed floating in the air, with the candles burning ().   But in our environment the flames go upward because the hot air is less dense, and that effect should disappear when no gravity is present, am I right? How would a candle flame look then? How do I automatically remap buttons on my mouse at startup? How to create a new figure in MATLAB? Nginx installation error in Ubuntu 16.04 :~$ sudo apt-get remove nginx* && sudo apt-get install nginx-full*  Reading package lists... Done Building dependency tree        Reading state information... Done  After this operation, 5,579 kB of additional disk space will be used. Do you want to continue? [Y/n] y Get:1 http://in.archive.ubuntu.com/ubuntu xenial-updates/universe amd64 nginx-full-dbg amd64 1.10.0-0ubuntu0.16.04.1 [3,789 kB] Fetched 3,789 kB in 33s (112 kB/s)                                              Selecting previously unselected package nginx-common. (Reading database ... 189767 files and directories currently installed.) Preparing to unpack .../nginx-common_1.10.0-0ubuntu0.16.04.1_all.deb ... Unpacking nginx-common (1.10.0-0ubuntu0.16.04.1) ... Selecting previously unselected package nginx-full. Preparing to unpack .../nginx-full_1.10.0-0ubuntu0.16.04.1_amd64.deb ... Unpacking nginx-full (1.10.0-0ubuntu0.16.04.1) ... Selecting previously unselected package nginx-full-dbg. Preparing to unpack .../nginx-full-dbg_1.10.0-0ubuntu0.16.04.1_amd64.deb ... Unpacking nginx-full-dbg (1.10.0-0ubuntu0.16.04.1) ... Processing triggers for ufw (0.35-0ubuntu1) ... Rules updated for profile 'Apache Full' Rules updated for profile 'Nginx HTTP'  Processing triggers for systemd (229-4ubuntu4) ... Processing triggers for ureadahead (0.100.0-19) ... Processing triggers for man-db (2.7.5-1) ... Setting up nginx-common (1.10.0-0ubuntu0.16.04.1) ... Setting up nginx-full (1.10.0-0ubuntu0.16.04.1) ... Job for nginx.service failed because the control process exited with error code. 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How do i select such questions only ? Is "You are correct" correct? Theoretically if I were able to build some sort of device that let me sit 1 foot away from the surface of the Sun (or any star for that matter) without being vaporized, would a star produce any sort of audible sound at that distance?   From my understanding, sound waves need a medium to propagate through (air, water, etc).   Does this mean that all stars are completely silent (in the audible spectrum of frequencies at least) as they fuse  hydrogen into helium?   I know stars emit radio, xray, infrared, ultraviolet electromagnetic waves, but how about audible (20 Hz to 20Khz) frequencies?   For some reason it seems like this process would produce some sort of sound.