The problem:     Prove that the only integer solutions of the equation $x^3=y^2+2 $ are $x=3$ and $y=\pm 5 $.   I found these related posts:       I have tried techinques similar to the ones used in those posts, however I have not been able to solve it yet.    Edit: The first comment gave me the hint of working in $\mathbb{Z}[\sqrt{-2}]$.  $$x^3=y^2-(-2)=(y+\sqrt{-2})(y-\sqrt{-2}) $$ And we define $a:=y+\sqrt{-2}$ and $b:=y-\sqrt{-2} $. Now I need to prove that $a$ is a cube in $\mathbb{Z}[\sqrt{-2}]$. As a side note, that would imply that $b$ is also a cube in $\mathbb{Z}[\sqrt{-2}]$. Once I prove that, I can use the answers in the post titled "", and solve the problem.  How can I prove that $a$ is a cube in $\mathbb{Z}[\sqrt{-2}]$ ?
Let $A =  \mathbb{Z}[\sqrt{-2}]= \{a+b\sqrt{-2} \ : a, b \in \mathbb{Z}\}$. Show that the only natural number $x$ for which $x+\sqrt{-2}$ is a cube in $A$ is $x=5$.   So I have to show that there exists $c,d \in \mathbb{Z}$ such that $(x+\sqrt{-2})^3= c + d \sqrt{-2}$. From the last equation, I obtain $x(x^2-6) = c \in \mathbb{Z}$ and $3x^2-2= b \in \mathbb{Z}$, and here I blocked. Could I conclude from both equations? If not, is there exists another way to proceed (hint)?  Thanks!
How distorted does the Andromeda Galaxy appear to us due to the speed of light? The Andromeda Galaxy appears to us at an angle to the galactic disk, i.e. we are not in the Andromeda Galaxy disc's plane, nor are we near the direction that the galaxy's pole points. Therefore, due to the geometry and distances involved, it would seem to me that we are seeing the 'far edge' a few thousand years later than we are seeing the 'near edge'. How far can a galaxy spin during that those few thousand years, and therefore how distorted are we seeing Andromeda? Can we infer how Andromeda or other large spiral galaxies seen at an angle would look if light were to travel instantly?  Might this phenomenon account for some of the unusual spin properties measured in galaxies, which is attributed to dark matter? I had this problem a week or two ago and I posted here but nothing from the suggested solutions seemed to work. I have managed to get some more details on it, in case anyone can help me, because this issue is causing me huge productivity problems and I really want it solved.  So I'm using Linux mint distribution, and I've installed Lyx via the software manager. Every time that I open a new document, I get an error claiming that some files are missing, and until they are installed, compiling the files or exporting them as pdfs is not possible. The file in question is called "extarticle.cls". Additionally, under Document->Settings->Document Class, every document class is marked as unavailable.  Here is the thing though.. this file is not missing... I have located it inside a folder associated with Lyx. It's definitely not missing.  I really need help with this one guys... It's been weeks since the problem came up and I really just want to be rid of it.. Why does this thing not work? Any help? 'pyrcc5' is not recognized as an internal or external command How to prove $\bar{m}$ is a zero divisor in $\mathbb{Z}_n$ if and only if $m,n$ are not coprime Given a real positive sequence $\{a_n\}$ such that $\sum_{i=1}^n a_i$ converges.  Prove that there exists a real sequence $\{c_n\}$ monotonically increasing to $\infty$ such that $\sum_{i=1}^\infty a_nc_n$ converges.  What I have done:  I have written $$\sum_{i=1}^{N+1}a_ic_i=c_{N+1}\sum_{i=1}^{N+1}a_i-\sum_{i=1}^N(c_{n+1}-c_n)S_n$$  and then try to prove the convergence by the Cauchy criterion. However my evaluation is still not good enough. I'm really stuck. Why are place names in some old books a capital letter followed by dashes, and what is this phenomenon called? This is something very odd I have noticed in some older books, most recently in "Crime and Punishment" by Dostoyevsky.  Place names, when not specifically essential or well-known, are made up of one letter followed by a line of underscores (i.e. "X_____").  I think I recall reading something similar in another older work, possibly by Jane Austin?  Apologies if this is a duplicate; I've searched quite extensively and can't seem to figure it out.  I don't know a name for this, so can't search easily for it. sudo: /var/lib/sudo writable by non-owner (040777), should be mode 0700 I get error message always when I work with sudo command.  It shows always sudo: /var/lib/sudo writable by non-owner (040777), should be mode 0700.   A while ago I changed the permission of /var folder to 777.   How can I fix it? Transfer from JFK Terminal 2 to Terminal 4 I have family who does not speak English.  They are changing planes in JFK Airport and will most likely be transferring from Terminal 2 (Delta) to Terminal 4 (China Southern Airlines).  At first I thought that directing them to use the Airtrain would be best.  However, in doing more reading my understanding is that the Airtrain would mean they would need to go through security again.  I found this on Delta's site:   If they use the JITNEY Shuttle to change from Terminal 2 to Terminal 4, does that mean they would not need to re-clear Security?  Edit: They are flying TO China. They are going from Tampa -> NY JFK -> China. How can I find the method that called the current method? Difference between cp -r and cp -R (copy command) I have some questions regarding suspended users:   Why do suspended users have a reputation of one? According to the default reputation of a new user, he/she will have reputation of one and he/she has the privilege of posting questions. When comparing this, this is getting violated when compared to suspended users' reputation. Even though they have one reputation point, they cannot post questions. As suspended users do not have privilege of posting questions, there is no use of making "Ask Question" visible. This is an unnecessary server hit. Am I right? As suspended users do not have the privilege of answering questions, there is no use of making "Answer Question" visible. Prime spectrums bijection but not homeomorphism How to install Eclipse C/C++ in Ubuntu 18.04 To clean install Eclipse for C/C++ on Ubuntu 18.04 I tried sudo apt install eclipse eclipse-cdt. It's showing an error, see the log file....  log file..... !SESSION Fri Aug 16 14:39:20 IST 2019 ------------------------------------------ !ENTRY org.eclipse.equinox.launcher 4 0 2019-08-16 14:39:20.065 !MESSAGE Exception launching the Eclipse Platform: !STACK java.lang.ClassNotFoundException: org.eclipse.core.runtime.adaptor.EclipseStarter     at java.base/java.net.URLClassLoader.findClass(URLClassLoader.java:471)     at java.base/java.lang.ClassLoader.loadClass(ClassLoader.java:588)     at java.base/java.lang.ClassLoader.loadClass(ClassLoader.java:521)     at org.eclipse.equinox.launcher.Main.invokeFramework(Main.java:626)     at org.eclipse.equinox.launcher.Main.basicRun(Main.java:584)     at org.eclipse.equinox.launcher.Main.run(Main.java:1438)     at org.eclipse.equinox.launcher.Main.main(Main.java:1414) 50 question per month limit? I get a message:     Sorry, you are only allowed to ask 50 questions in a 30 day period   Why is there a question limit? 50/month is just not enough :) How to get a cross-origin resource sharing (CORS) post request working "Over vs Above" and "Under vs below" Is is possible to say that "Above" means "over" and "Under" means"below", but not the opposite?  According to my grammar:  Over - In a higher position (close or touching) Above - In a higher position  Under - In a lower position (close or touching) Below - In a lower position  Example:     There's a helicopter flying round above / over the school.   I would interpret this sentence as "above" meaning almost touching the school or not. And If I hear "over", the helicopter is almost touching school.  This makes me think about "over" and "on". "On" means that something is touching something, but if "over" can mean also "touch".Well, I would never use "over" to mean "touching something", I would use "on" instead. It seems that Mac OS X 10.6.1 Snow Leopard has a tendency to leave some windows off-screen at times, especially when I disconnect an external monitor from my MacBook. How can I move the window back on-screen when it's not possible to grab the title bar and drag it back onto the screen?   recommends adjusting your screen resolution downward to have off-screen windows "snap back into view" and then adjust your screen resolution back to its original setting. In OS X 10.4 Tiger, I had a script that brought all off-screen windows back on-screen, but in my upgrades to Leopard and now Snow Leopard, I don't seem to have that script.  Any better solutions out there? How many non congruent trapeziums are possible with the side lengths from {1,3,4,5,6} I need to find number of mutually non  congruent trapeziums with side lengths from {1,3,4,5,6}.   I know for a trapezium with a,c as non parallel sides the condition for my question is that  $$|a-c|&lt;b+d&lt;a+c$$  But i cant apply it to count the cases. Can someone help in how to get the number of cases? the relationship between maximizing the likelihood and minimizing the cross-entropy Where should I start with Star Trek? (TV series only, not the movies) I haven't seen any of the stuff in the Star Trek franchise, except for one of the movies, and now I want to remedy the situation. I am not interested in the movies, just the TV series. However, since the show has many seasons, I am not interested in watching the show start to finish. So, where (i.e what series/season) should I start with? Do I really need to see the series in real-world chronological order?   Note: I am somewhat familiar with Star Trek, that is I know the basic premise - ships in space, with war between different space-faring sentient species.