Problem :   Let $n,k$ be natural numbers   Prove that there no $k$ such that :      $$3n^{2}+3n+7=k^{3}$$    I see this short solution from a book :   We use $\pmod{9}$   $k^{3}\equiv 0,1,8\pmod{9}$   And $3n^{2}+3n+7\equiv 4,7\pmod{9}$   $\implies $ no solution ?   But I don't understand why he chose mod $9$ and how we know that $k^{3}\equiv 0,1,8\pmod{9}$ ?   And how he know $3n^{2}+3n+7\equiv 4,7\pmod{9}$ ? Can someone explain to me or give me another solution !
I have this really hard problem that I am working on and I just don't seem to get it. The question is: let $n$ be a positive integer; prove that there does not exist any integer $m$ such that $3n^2+3n+7=m^3$.  Please explain in detail otherwise I might not understand. I am also open for any methods that might help. Thank you. Please help.
Is there a word for someone who has others do all their work for them? In a way like a slave master. More though like say someone who had robots do all their work and they were never themselves productive and had no drive. If you've read R.U.R. you'll understand the question better. I'm looking for a word with more definition than just lazy.  Update 1: It think I may actually go with drone if anything. I think that definition fits what I'm looking for the best. I also liked the definition of the British term layabout that one person mentioned, but I will probably use drone. Thanks for all the help and sorry if I was confusing or unclear.  Update 2: I needed this word for use in an essay I was writing which I have now handed in. Thanks for the help guys. No more suggestions needed. Get last result in interactive Python shell would this be considered a 'methodology' or a 'method'? Say I developed something called "Methodology for Establishing Remaining Life of Components". Basically, at work, we have components (or equipment). I wanted to develop a way to identify how much longer the components / equipment can last before they ware out and cannot be used.  It takes into account "the degree of damage currently in the component, the rate of future damage accumulation based on realistic operating scenarios, and the degree of damage required to cause failure." Taking these three into account, the remaining life of a component can be assessed. This is what I call "Methodology for Establishing Remaining Life of Components".   My question is, would this be considered a 'methodology for establishing remaining life of components' or just a 'method for establishing remaining life of components'? Or do you need more information to figure out if it is a method or methodology?   I cannot provide all the information for business purposes, but if additional  information is required to figure out if it is a method or methodology, I will provide it. I use the grep command to search for a string in many files. Is there something similar to "search and replace"? Add table row in jQuery Initializing K-means centers by the means of random subsamples of the dataset? If I have a certain dataset, how smart would it be to initialize cluster centers using means of random samples of that dataset?  For example, suppose I want 5 clusters. I take 5 random samples of say, size=20% of the original dataset. Could I then take the mean of each of these 5 random samples and use those means as my 5 initial cluster centers? I don't know where I read this but I wanted to know what you guys think about the idea.    UPDATE: Please see this thread  for the general discussion about the various initialization methods. I wanted to try out different desktop environments so I installed XFCE, KDE, GNOME, etc on top of Unity in Ubuntu 12.04  After a while I decided that I didn't like those other DEs and would stick to Unity. So I changed my default DE to Unity by configuring gdm.  sudo dpkg-reconfigure gdm   Now I am able to choose my DE at login time and all of DEs are working properly.   But the strange thing is that my boot up screen says 'lubuntu', my login screen is KDE, and my desktop is Unity.   How and why is this happening? Why didn't my gdm configuration have any effect? Does login and boot up screen need to be configured separately from the DE?  There are other similar questions here but they are not the same as this one. I do not want to remove the other enviroments I'm quite happy with having a list of DEs to select from. I want to know how to set proper defaults. How to check internet access on Android? InetAddress never times out appears to have been question banned because of two unpopular Meta suggestions. (He says so in )  Look at the two contributions that appear to have gotten him banned. Like those of many others, they may not be of perfect quality, but they are not actively harmful to the site, either. As it stands, you can get question blocked on Meta simply for asking questions that aren't popular with the Meta crowd.  Why does the team insist on keeping the question ban active on Meta?  Why can't real problem users be dealt with through manual moderator action? Don't tell me it's a problem of scale; as far as I can see, actually toxic users are very rare on Meta.  People will frequently complain that Meta actively discourages the expression of unpopular viewpoints and suggestions. With the threat of question bans in place, I find those complaints difficult to refute. I hate that. Is the expression      All women who are mothers of daughters are here.   correct? I mean every woman who has at least one daughter is here. I have joined corresponding Local Enterprise Partnerships table data into a polygon shapefile containing UK Local Authorities.      How do I Dissolve or Merge the the Local Authorities polygons by LEP_NAME as shown in the screenshot?   Ideally, I would like to retain the Local Authority attributes with the resultant merged polygons. I have zero scripting knowledge so am looking for a tool based solution. Any help or pointers much appreciated.    Additional Note: When attempting to use the dissolve tool the LEP_NAME field is not available as shown above. The LEP attribute data was created in Excel, saved as CSV format and joined to the Local Authority polygon shapefile. There are duplicate IDENTIFI0 fields as shown in the other screenshot - could this be the root of the issue? I'm trying to prove : Given a simple graph $G$ with $n$ vertices, where $n$ is even, prove that if every vertex has degree $\dfrac n2 + 1$, then $G$ must contain a (simple) $3$-cycle. A (simple) $3$-cycle is a set of $3$ (distinct) vertices, $a, b, c$ such that $ab$ is an edge, $bc$ is an edge and $ac$ is an edge.  I know there are similar ones already on here, but I can't comprehend any of the answers. This is what I have so far:  Base Case: Since the graph has to have at least $3$ vertices to have a simple $3$-cycle, the lowest possible even number of vertices is $n = 4$.   $\dfrac42 + 1 = 3$ degree (edges attached to node). This means each vertex connects to each of the other $3$ vertices since $G$ is a simple graph. Thus, any combination of $3$ vertices will be a simple $3$-cycle.  Inductive Hypothesis: Assume a simple graph with $k$ vertices, where $k$ is even, contains a simple $3$-cycle if each vertex has degree $\dfrac k2 + 1$.  Inductive Step: Since only concerned about when the graph has an even number of vertices, going to show a simple graph with $k + 2$ vertices, where $k$ is even, contains a simple $3$-cycle if every vertex has degree $\dfrac{k+2}{2} + 1$   This is as far as I get and don't know how to finish the Inductive Step (Also please feel free to let me know if I messed anything up, up to this point) Filesystem owner changed by mistake Defining new unit (year, century) in siunitx I'm having a hard time figuring out the LaTeX commands to define new units of year and century in siunitx.  I'd like to write \SI{3}{\cm\per\year} and have it show: 3 cm yr-1, or \SI{42}{\micro\meter\per\century} become 4.2E-5 m/century Close Vs Closed and Open Vs. Opened Which board is proper to describe the state of any shop/restaurant?  For instance,     A restaurant with the board Close OR Closed?    A restaurant with the board Open OR Opened?   Well, if I think 'closed' is used when the restaurant is permanently or temporarily closed for more than a day (say for holidays or renovation), what about the conversation mentioned below? It takes -ed even for that day!     "Hey Mike, I forgot my specs at home. Do you see the restaurant there at distance? Is it open?   "Yeah, I do see. But it's closed.   I think open/opened or close/closed here is not a verb. If it isn't, then what is it in those examples? how to bring up the menu at the bottom left hello everyone I have a little problem, how do I bring up the menu in the circle? when I select another object the menu becomes non-existent I have a class like this:  public class myClass {   public List&lt;myOtherClass&gt; anewlist = new List&lt;myOtherClass&gt;;    public void addToList(myOtherClass tmp)   {     anewList.Add(tmp);   }  }   So I call "addToList" a hundred times, each adding a unique item to the list.  I've tested my items to show that before I run the "addToList" method, they are unique.  I even put a line in to test "tmp" to make sure it was what I was expecting.  However, when I do this (lets say myClass object is called tmpClass):  int i = tmpClass.anewList.Count(); for (int j = 0; j&lt;i; j++) {    //write out each member of the list based on index j... }   I get the same exact item, and it's the last one that was written into my list.  It's as if when I add, I'm overwriting the entire list with the last item I've added.    Help?  This makes no sense. I've also tried List.Insert, where I'm always inserting at the end or at index 0.  Still no dice.  Yes, I'm doubly source my indexing is correct and when I do my test I'm indexing through each of the elements.  :)  UPDATE: Okay, I tried this and still had the same problem:  foreach(myOtherClass tmpC in tmpClass.anewList) {         Console.WriteLine(tmpC.theStringInMyClass.ToString()); }   and still for each of the 100 items, I got the same string output... I'm sure I'm doing something completely stupid, but I don't know what yet. I'm still 100% sure that the right string is getting passed in to begin with.  -Adeena    Okay, I tried this and still had the same problem:  foreach(myOtherClass tmpC in tmpClass.anewList) {     Console.WriteLine(tmpC.theStringInMyClass.ToString()); }   and still for each of the 100 items, I got the same string output...  I'm sure I'm doing something completely stupid, but I don't know what yet. I'm still 100% sure that the right string is getting passed in to begin with.  -Adeena Applying MVT and IVT (?) to show something about f'''? (edit: actually, Taylor's Theorem using Lagrange remainder) I believe that the correct answer to this question will involve applying both the Mean Value Theorem and the Intermediate Value Theorem, perhaps several times.  However, I can't see how to proceed.  I went down what I thought were the typical paths and just reached dead ends.  Does someone maybe have a hint?  I think that with the appropriate nudge I should be able to solve this question.  If not, I'll edit and ask if someone knows how to solve the whole thing.  Here's the question:  Let $f: \mathbf R \to \mathbf R$ be such that $f$, $f'$, $f''$, and $f'''$ exist and are continuous on $\mathbf R$ and satisfying $f(-3)=-1$, $f(0)=0=f'(0)$, and $f(3)=8$.  Prove that there exists $\xi \in (-3, 3)$ such that $f'''(\xi) \ge 1$. Negative pressure How is negative pressure created in a fluid system?Isn't it counter intuitive that we are reducing the pressure of a system containing no molecules(zero pressure) to a lower value? Advanced statistics books recommendation There are several threads on this site for book recommendations on  and  but I am looking for a text on advanced statistics including, in order of priority: maximum likelihood, generalized linear models, principal component analysis, non-linear models. I've tried  by A.C. Davison but frankly I had to put it down after 2 chapters. The text is encyclopedic in its coverage and mathematical treats but, as a practitioner, I like to approach subjects by understanding the intuition first, and then delve into the mathematical background.   These are some texts that I consider outstanding for their pedagogical value. I would like to find an equivalent for the more advanced subjects I mentioned.   , D. Freedman, R. Pisani, R. Purves. , R. Hyndman et al. , T. Z. Keith , Rand R. Wilcox , Gareth James, Daniela Witten, Trevor Hastie and Robert Tibshirani , Hastie, Tibshirani and Friedman (2009)