Prove that every integer is a sum of two squares and a cube of integer Prove that every integer is a sum of two squares and a cube of integer.  Probably the only connected result I know, is the Fermat's theorem on sums of two squares. But I don't know how to prove that in other cases (for numbers not being of the form $x^2+y^2=x^2+y^2+0^3$) one can choose such two squares that the remainder will be a cube.
How prove this diophantine equation $x^2+y^2+z^3=n$ always have integer solution show that: For any postive ineteger $n$,then the equation    $$n=x^2+y^2+z^3$$   always have integer solution   My idea: such as $n=1$,then we have $$1=0^2+0^2+1^3$$ $$2=0^2+1^2+1^3$$ $$3=1^2+1^2+1^3$$ $$4=2^2+0^2+0^3$$ $$5=1^2+2^2+0^3$$ $$6=1^2+2^2+1^3$$ $$7=2^2+2^2+(-1)^3$$ $$8=0^2+0^2+2^3$$ $$9=1^2+0^2+2^3$$ $$10=1^2+1^2+2^3$$ $\cdots\cdots\cdots$  But for  general $n$, How prove it?
Dual boot windows 8 and Ubuntu with Windows 8 Boot manager I have two partitions on my hard-didk , I have installed ubuntu on my 1st partition and windows 8 later on another partition.Now i can only boot into windows 8 because it doesn't recognize Ubuntu.  How would i dual boot my PC without using grub . I would like using Windows 8 boot manager as its pretty neat.  This is what i have tried :  I tried using easyBCD but it doesn't work.It causes the boot manager to switch to windows 7 Boot Manager .  EasyBCD creates a C:/NST/AutoNeoGrub0.mbr file .  As EasyBCD was causing the boot manager to switch i created a manual entry to the BCD store uisng windows bcdedit command and point it to the C:/NST/AutoNeoGrub0.mbr file created by EasyBCD. This gave me the Windows 8 Boot-Manager with Ubuntu Entry,but grub is unable to find the root partition.  Is there any other work around or solution for this ?  Any help would be appreciated.  Note: The windows 8 boot Manager is sky blue color interactive menu with mouse and other options and windows 7 boot manager is the normal black and white one where you can only use your keyboard  I don't want to install grub because it would prevent my windows 8 from updating as its a genuine MSDN version and various other reasons We all know very well how to make an angle of 15 and 45 degree and its multiple using compass. Can anybody tell me how to make an angle of 10 degree and its multiple using compass? $1+x+\ldots+x^n$ perfect square Let $p$ be the polynomial $p(x)=1+x+\ldots+x^n$. For which couples $(a, n)\in\mathbb{N}^2$, $p(a)$ is a perfect square? I'm particularly interested in $p(3)$. What are the rules for China's visa-free transit programs? Assuming one wants to travel through China, what are the rules for visa-free access? To be more specific:   How does the 24-hour visa free transit work? How does the 72-hour version work? What does a person ineligible for either of these programs have to do to transit China? Do the programs apply to flights only or do they also work for transit on land? Would a visitor save much hassle by using visa free transit instead of just getting a visa?      NB: this is intended as a canonical question on transiting China You require a reputation of 50 to leave comments on others' questions/answers; the FAQ simply states this as a fact but does not explain it. I'm a new user and I have identified this as an issue very quickly through my own experiences so far. Looking at other questions seems to also provide justification against this 50 reputation requirement for comments on others' questions/answers.      In my own experience, I've seen a few questions like "how do I do x?" then the same poster has answered with "in the end I did it like this: ..." where their solution was perfectly sensible, but there may be some gotchas to watch for - in this case I want to vote up their answer and/or comment on it to point out the potential gotchas, but I can do neither!  To me all of the above seems justification against this rep requirement to comment, so what is the counter-argument and justification for it? Accepted answer appearing below top voted On , sorted by votes, the accepted answer is #3 in order of votes.  It appears as number 3 in the list as well. Is this new behavior intended? How do I fix this "E: Type '*' is not known on line * in source list ..." update error? Every once in a while I see users having issues to update due to errors like this:  E:Type 'ain' is not known on line 1 in source list /etc/apt/sources.list.d/some-ppa.list'   with varying types/line numbers/source list files (often after removing a PPA).  How can such an error be fixed? Binding single key (such as 'F1') to multiple keys (such as 'Ctrl+C') in Gnome I know how to map a single key to a command using the gconf settings. I also know how to map a single key to another key (F1 -> ctrl) using Xmodmap. My question is how can I map a single key such as F1 to a multi-key operation such at ctrl+c. I am basically trying to set up a single key to simulate copy. I can think of two ways to possibly do this:   Map single key to multiple keys so that when the single key is pressed it is essentially pressing ctrl+c. Use gconf settings to execute the copy “command” upon the single key press. I am not sure if there is a “command” to copy text. Why does my object not show up? How to use Class in Java? There's a good discussion of Generics and what they really do behind the scenes over at , so we all know that Vector&lt;int[]&gt; is a vector of integer arrays, and HashTable&lt;String, Person&gt; is a table of whose keys are strings and values Persons.  However, what stumps me is the usage of Class&lt;&gt;.  The java class Class is supposed to also take a template name, (or so I'm being told by the yellow underline in eclipse).  I don't understand what I should put in there.  The whole point of the Class object is when you don't fully have the information about an object, for reflection and such.  Why does it make me specify which class the Class object will hold?  I clearly don't know, or I wouldn't be using the Class object, I would use the specific one. If $\sum_{n_0}^{\infty} a_n$ diverges prove that $\sum_{n_0}^{\infty} \frac{a_n}{a_1+a_2+...+a_n} = +\infty $ I've tried to use the limit test: $\lim_{n \rightarrow \infty} \frac{a_n}{b_n} = C$ in the series of $a_n$ and $\frac{a_n}{a_1+a_2+...+a_n}$ but it was inconclusive.  The question does not especify the sign of $a_n$ but proving it for positives would already help.  I think the sign can change as long as the partial sums never get to be zero. Like if the series has only negative numbers the result seems to stil hold. No indent in the first paragraph in a section? Finite subgroups of the multiplicative group of a field are cyclic In Grove's book Algebra, Proposition 3.7 at page 94 is the following     If $G$ is a finite subgroup of the multiplicative group $F^*$ of a field $F$,   then $G$ is cyclic.   He starts the proof by saying "Since $G$ is the direct product of its Sylow subgroups ...". But this is only true if the Sylow subgroups of $G$ are all normal. How do we know this? Krull dimension of this local ring I want to know what the Krull dimension of this ring $\mathbb C[x,y]_p/(y^2-x^7,y^5-x^3)$ is, where $p\neq (0,0)$. I know the dimension of it in the origin point, but I don't know other cases. I have 2 columns in Excel: the first contains a list of employee names who perform function A, and the second contains a list of employee names who perform function B.  What I want is to identify, in one cell, if there is one employee whose name appears in both the first and second columns. It does not have to show which name(s) appears in both columns, it just needs to identify IF there are any names that appear in both columns. How to write (La)TeX (with parentheses) [or any other TeX-related logo] Why are the rooting apps not working anymore? A couple years ago, it was easy to root any device: Install KingRoot, KingoRoot, or a similar app, hit the large ROOT button, and let it do the rest. After 5 minutes or so, a green "Root Succeeded!" prompt will pop up, and we can choose to keep the root app, or uninstall it and install a clean root manager app like SuperSU/Superuser.  How do they work? Why aren't they working anymore? And if they work, should I use them or not? Find all functions $f(x)$ such that $f\left(x^2+f(y)\right)$=$y+(f(x))^2$ Let $\mathbb R$ denote the set of all real numbers. Find all function $f: R\to \ R$ such that   $$f\left(x^2+f(y)\right)=y+(f(x))^2$$   It is the problem. I tried to it by putting many at the place of $x$ and $y$ but I can't proceed. Please somebody help me. In Bishop's book "Pattern Classification and Machine Learning", it describes a technique for regularization in the context of neural networks. However, I don't understand a paragraph describing that during the training process, the number of degrees of freedom increases along with the model complexity. The relevant quote is the following:     An alternative to regularization as a way of controlling the effective   complexity of a network is the procedure of early stopping. The   training of nonlinear network models corresponds to an iterative   reduction of the error function defined with respect to a set of   training data. For many of the optimization algorithms used for   network training, such as conjugate gradients, the error is a   nonincreasing function of the iteration index. However, the error   measured with respect to independent data, generally called a   validation set, often shows a decrease at first, followed by an increase as the network starts to over-fit. Training can therefore be   stopped at the point of smallest error with respect to the validation   data set, as indicated in Figure 5.12, in order to obtain a network   having good generalization performance. The behaviour of the network   in this case is sometimes explained qualitatively in terms of the   effective number of degrees of freedom in the network, in which this   number starts out small and then to grows during the training process,   corresponding to a steady increase in the effective complexity of the   model.   It also says that the number of parameters grows during the course of training. I was assuming that by "parameters", it refers to the number of weights controlled by the network's hidden units. Maybe I'm wrong because the weights are prevented to increase in magnitude by the regularization process but they don't change in number. Could it be referring to the process of finding a good number of hidden units?  What's a degree of freedom in a neural network? What parameters increase during  training? I am building a drupal site using Views, search API module and user-points module.  I created the "database server", and indexed many fields of nodes.  I created a search page using views and the search API, that present content (image, title etc.) In addition, I want to add to every content in this search view the name of the author that wrote it, his author picture.  I add the Indexed Content: Author relationship. Unfortunately, I can't find a way to add the author picture and the current total point to my view.  Can somebody advice how to do that?