So, I'm having trouble doing this certain problem. As the title suggests I should prove that if this holds: $$x | y^2 $$ $$y^2 | x^3$$ $$x^3 | y^4$$ $$...$$ Than for each natural x and y, x is equal to y. I've tried writing the numbers x and y in form of: $$x = p_1^{\alpha_1} + p_2^{\alpha_2} + ... + a_n^{\alpha_n}$$ $$y = P_1^{\beta_1} + p_2^{\beta_2} + ... + p_n^{\beta_n}$$ Now when I try to express the given equations in this form I get: $$x \le y^2 \le x^3 \le y^4$$ Which for each $i$ yields: $$\alpha_i \le 2\beta_i \le 3\alpha_i \le 4\beta_i ...$$ I was now wondering what to do next to prove it. Thanks.
After I saw that post i wanted to solve the first one which is $a\mid b^2,b^2\mid a^3,a^3\mid b^4,b^4\mid a^5\cdots$ Prove that $a=b$  Now i started by proving that $a$ and $b$ have same prime divisors,proving that is trivial,after doing so I checked do those factors have to have the same power.I tried with $$a=p^2,b=p^3$$ I noticed it exactly goes for 3 terms,or that $a^3\mid b^4$ doing $$a=p^{n-1},b=p^n\\p^{n-1}\mid p^{2n}\\p^{2n}\mid p^{3n-3}\\p^{3n-3}\mid p^{4n}\\p^{4n}\mid p^{5n-5}\\\cdots\\p^{n(n-1)}\mid p^{n(n-1)}\\p^{n(n-1)}\mid p^{n(n+1)}\\p^{n(n+1)}\not\mid p^{(n-1)(n+1)}$$ Now yeah I guess that would be a proof,but wouldn't setting $n=n+1$ infinitely many times make every term dividable?
Why are we still using CPUs instead of GPUs? It seems to me that these days lots of calculations are done on the GPU. Obviously graphics are done there, but using CUDA and the like, AI, hashing algorithms (think bitcoins) and others are also done on the GPU. Why can't we just get rid of the CPU and use the GPU on its own? What makes the GPU so much faster than the CPU? Favourites Page on iOS App I would like to have a page listing all the questions I marked as favourite, same as the .  Also, I think the desktop version, the star, is much more intuitive and easier to find. Thanks! User is able to edit a file using vim I have a file named file.txt with some contents located in /home/myuser directory and I change the permissions of that file to root.root and 644.   -rw-r--r-- 1 root root 1.4K Jan 15 07:09 file.txt   Then I switched to myuser and opened it with vim(without sudo) and tried to edit and save it :wq! and I got output like:  "file.txt" E949: File changed while writing Press ENTER or type command to continue   Then I hit enter and then again :wq! and the output was :  WARNING: The file has been changed since reading it!!! Do you really want to write to it (y/n)?   I hit y, the file got saved and exit from vim.  The question is why a file with ROOT privileges and read-only access for a user changed by the way?  PS: 1. myuser is a sudo user.   The user permissions after saving the file changed to myuser.myuser as well.  -rw-r--r-- 1 myuser myuser 1.4K Jun 27 18:57 file.txt What is the correspondence between the conserved canonical energy-momentum tensor, which is $$ T^{\mu\nu}_{can} := \sum_{i=1}^N\frac{\delta\mathcal{L}_{Matter}}{\delta(\partial_\mu f_i)}\partial^\nu f_i - \eta^{\mu\nu}\mathcal{L}$$ (the four conserved Noether currents corresponding to four possible spacetime translations)  where $\{f_i\}_{i=1}^N$ are the $N$ matter fields in the theory, and we assume $f_i\mapsto\alpha^\nu\partial_\nu f_i$ for translations,  and stress-energy tensor from the Einstein-Hilbert action, which is: $$ T_{\mu\nu}\equiv-\frac{2}{\sqrt{-g}}\frac{\delta\mathcal{L}_{Matter}}{\delta g^{\mu\nu}} $$   In particular, how do you get that the two are equal (are they?) for Minkowski space, for which there is no variation in the metric? The Safari Back Button Problem I do some minor programming and web work for a local community college. Work that includes maintaining a very large and soul destroying website that consists of a hodge podge of VBScript, javascript, Dreamweaver generated cruft and a collection of add-ons that various conmen have convinced them to buy over the years.   A few days ago I got a call "The website is locking up for people using Safari!" Okay, step one download Safari(v3.1.2), step two surf to the site. Everything appears to work fine.  Long story short I finally isolated the problem and it relates to Safari's back button. The website uses a fancy-pants javascript menu that works in every browser I've tried including Safari, the first time around. But in Safari if you follow a link off the page and then hit the back button the menu no longer works.  I made a pared down webpage to illustrate the principle.  &lt;!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN"&gt; &lt;html&gt; &lt;head&gt;&lt;title&gt;Safari Back Button Test&lt;/title&gt; &lt;meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1"&gt; &lt;/head&gt; &lt;body onload="alert('Hello');"&gt; &lt;a href="http://www.codinghorror.com"&gt;Coding Horror&lt;/a&gt; &lt;/body&gt; &lt;/html&gt;   Load the page and you see the alert box. Then follow the link off the page and hit the back button. In IE and Firefox you see the alert box again, in Safari you do not.  After a vigorous googling I've discovered others with similar problems but no really satisfactory answers. So my question is how can I make my pages work the same way in Safari after the user hits the back button as they do in other browsers?  If this is a stupid question please be gentle, javascript is somewhat new to me. How much creative independence are the Show runners accorded in Game of Thrones? Since Season 3, we have been observing that the show was taking a different direction than the books. Many important characters never made it into the show, others who are still alive were killed off.   As we know the show is supposed to be based on the books, not to be direct adaption of the books from print to television media, there's bound to be a certain degree of creative independence accorded to them by the author of the series, George R.R. Martin.   The question is, how much independence do they have? Do they still have to take GRRM on board with whatever changes they might intend to make? By "they" I mean D&amp;D and the whole writers crew for the show. Given the number of significant changes they have made, I reckon the answer would be a lot. Has GRRM or HBO ever commented on that? Create model from XYZ data points I'm trying to create a 3D heat map model out of data that I have from a csv file. I have a few thousand data points with XYZ coordinates.  I'm able to import the data into blender as a point cloud with , however I would like to create a model with solid faces. Is there a way when importing the data to somehow join the points to create faces instead of generating thousands of separate points?  Currently, this what I have in terms of point cloud  The end product would look something like this (minus the texture) and obviously the model wouldn't be as regular as this one  Any help is greatly appreciated! Why do some badges appear white in color and others are dark? Creating a responsive menu system for a site with a lot of content I am currently creating a website which has a lot of content - text. For this I need a serious menu structure with 5-10 menu items per menu and at least 15 menu-blocks per main menu item.   Further it has to be accessible via mobile.   Till now I have used panels and added menus inside the panels. But mobile isn't to happy about that and the list-names in the menus overlap the content.   How would you go about solving this issue? I trust you experts here! :-) Why did the snake wink at Harry at the zoo? This is mostly a memory question - I'm having a hard time remembering this and I can't read the book right now.  On the first book, a snake at the zoo winks at Harry. For the sake of the question, we'll assume that snakes can in fact wink.  I have a fuzzy memory here - I think that it winked before Harry had any kind of interaction with the snake (was this before or after Harry broke the glass?)  So, why did the snake wink at Harry? Given that they had not met ever before.  I suppose that the only reason it would wink at Harry would be because it knows Harry can communicate with it. But this would imply that snakes are capable of sensing such ability on wizards - is there (other) evidence that this is true? Why is torque a cross product? If I'm not wrong, torque is perpendicular to both the radius and force i. e. It is along the axis of rotation. Questions that arise are- why do we consider the length between the axis/point of rotation while calculating torque? More importantly why is torque a cross product? What is the difference between using IDisposable vs a destructor in C#? When would I implement IDispose on a class as opposed to a destructor?  I read , but I'm still missing the point.    My assumption is that if I implement IDispose on an object, I can explicitly 'destruct' it as opposed to waiting for the garbage collector to do it.  Is this correct?  Does that mean I should always explicitly call Dispose on an object?  What are some common examples of this? Benefits of using QQ-plots over histograms In , Nick Cox wrote:     Binning into classes is an ancient method. While histograms can be useful, modern statistical software makes it easy as well as advisable to fit distributions to the raw data. Binning just throws away detail that is crucial in determining which distributions are plausible.   The context of this comment suggests using QQ-plots as an alternative means to evaluate the fit. The statement sounds very plausible, but I'd like to know about a reliable reference supporting this statement. Is there some paper which does a more thorough investigation of this fact, beyond a simple “well, this sounds obvious”? Any actual systematic comparisons of results or the likes?  I'd also like to see how far this benefit of QQ-plots over histograms can be stretched, to applications other than model fitting. Answers on  agree that “a QQ-plot […] just tells you that "something is wrong"”. I am thinking about using them as a tool to identify structure in observed data as compared to a null model and wonder whether there exist any established procedures to use QQ-plots (or their underlying data) to not only detect but also describe non-random structure in the observed data. References which include this direction would therefore be particularly useful. The main effect will be non-significant if the interaction is significant? Given that $a$~$\mathcal{N}(\mu, \Sigma)$ and $\mu$~$\mathcal{N}(0, \sigma^2 I)$, Is $p(a|\Sigma, \sigma)=\int p(a|\mu, \Sigma)p(\mu|\sigma)d\mu$ again a Normal distribution? And how to estimate the mean and variance respectively? Thanks a lot! How to monitor bandwidth use of each device on wifi network What will be the one's digit of the remainder in: $\left|5555^{2222} + 2222^{5555}\right|\div 7=?$ What will be the ones digit of the remainder in: $$\frac{\left|5555^{2222} + 2222^{5555}\right|} {7}$$ What is the best way to filter a Java Collection? I want to filter a java.util.Collection based on a predicate. I have a script that I need to execute on an NTFS partition. The script's permission is set to 600.  I attempted to modify the permissions by running chmod 755 script.sh, which doesn't report a failure or anything - but it also doesn't change the permissions on the file:  $ stat script.sh    File: `script.sh'   Size: 297070      Blocks: 584        IO Block: 4096   regular file Device: 811h/2065d  Inode: 35515       Links: 1 Access: (0600/-rw-------)  Uid: ( 1000/  xxxxxx)   Gid: ( 1000/  xxxxxx) Access: 2010-09-30 14:05:16.041621000 -0700 Modify: 2010-09-30 14:05:05.070157000 -0700 Change: 2010-09-30 14:05:05.070475000 -0700  $ chmod 755 script.sh $ stat script.sh    File: `script.sh'   Size: 297070      Blocks: 584        IO Block: 4096   regular file Device: 811h/2065d  Inode: 35515       Links: 1 Access: (0600/-rw-------)  Uid: ( 1000/  xxxxxx)   Gid: ( 1000/  xxxxxx) Access: 2010-09-30 14:05:16.041621000 -0700 Modify: 2010-09-30 14:05:05.070157000 -0700 Change: 2010-09-30 14:05:05.070475000 -0700   As you can see, it remains unchanged. Rotate Geographic map