To prove that means $n\neq x^2+y^2$, so I started with $n\equiv 3 (mod 4)$,  then I don't know how to go on.
Let $u$ be an integer of the form $4n+3$, where $n$ is a positive integer. Can we find integers $a$ and $b$ such that $u = a^2 + b^2$? If not, how to establish this for a fact?
I'm preparing a document using the book class. In the table of contents, the space between the subsection number and title is too wide. How to reduce it? Does ReLU layer work well for a shallow network? How do I calculate rotation caused by bounce friction? Suppose all the Associated Primes are Minimal. Let $R$ be a commutative Noetherian ring with unit and let $I$ be a fixed ideal. I am sorry if the following turns out to be a very silly question.  1) Suppose $\operatorname{Ass}(R/I)$ are all minimal, then what can we say about $R/I$?  One conclusion is that $I$ has a unique primary decomposition. If $R/I$ is CM, then $\operatorname{Ass}(R/I)$ are all minimal, but I don't think the converse is true. Can someone provide a counterexample?  2) Suppose $\operatorname{Ass}(R/I)$ are all minimal and have the same height. I still dont think we can conclude that $R/I$ is CM. Can someone provide a counterexample? Theorem 2.1.6 of Bruns and Herzog, CM Rings, gives a result in this direction but it needs additional hypothesis. It says that if every ideal $I$ is generated by height $I$ elements and $\operatorname{Ass}(R/I)$ are all minimal, then $R$ is CM.  It would be nice if someone can say something nice about the ring in the special situation described in 1) and 2).  Thanks. In Robert Dixon's Mathographics, a regular pentagon is constructed with straightedge and compass only.  It is the pentagon $ABCDE$ pictured below.   I am having trouble seeing why the central angles are all $72^\circ$, though. Can anyone provide the proof?  Also, does anyone know who this construction is due to? I haven't seen it anywhere, other than in Dixon's book; is it Dixon's result?     The result appears much more impressive without all the labels (which are slightly misplaced, please excuse this); however, I provided those so that answering would be easier. Also, it makes it easy to describe the steps in the construction:  1) Draw a circle (the red one)  with center $h$.  2) Draw the perpendicular lines $\ell_1$ and $\ell_2$ through $h$. Locate the points of intersection $f$, $B$, and $g$ with the red circle.  3) Bisect the line segment $gh$. Denote the center by $a$.  4) Draw the green circle with center $a$ and radius $ah$.  5) Draw the other green circle (as in (3) and 4)).  6) Draw the line segment through $f$ and $a$.  7) Locate the points of intersection $b$ and $c$ of the line segment with the circle constructed in step 4).  8) Draw the blue arcs (both have center at $f$ and the radii are $fb$ and $fc$).  9) Locate the points of intersection $A$, $C$, $D$, and $E$.   I actually have the solution to the first question, and will post it unless a more elegant explanation is provided (which is probably likely).  However, I find this construction particularly   beautiful, and would like to know who it is attributed to (Dixon doesn't say, explicitly). Scheduling a Batch job at a particular time I have a Apex batch class which I would like to schedule at a particular date time. The date time is stored in a variable and changes dynamically. ex Datatime scduleDateTime = datetime.now() ; I want to schedule the apex batch file to run at this datetime variable value.  How can I do that using system.scheduleBatch()  I started with this but could'nt see how I can specify a datetime variable in here:  datetime d = datetime.now(); sampleapexbatch b = new sampleapexbatch(d); System.scheduleBatch(b,'Sample Job',XXXXXXXX)   Any help is appreciated! Let $a$ and $b$ be two integers such that $\left(a,b\right) = 1$. Prove that $\left(a+b, ab\right) = 1$.   $(a,b)=1$ means $a$ and $b$ have no prime factors in common  $ab$ is simply the product of factors of $a$ and factors of $b$.  Let's say $k\mid a+b$ where $k$ is some factor of $a$.  Then $ka=a+b$ and $ka-a=b$ and $a(k-l)=b$.  So $a(k-l)=b, \  a\mid a(k-1)$ [$a$ divides the left hand side] therefore $a\mid b$ [the right hand side].  But $(a,b)=1$ so $a$ cannot divide $b$.  We have a similar argument for $b$.  So $a+b$ is not divisible by any factors of $ab$.  Therefore, $(a+b, ab)=1$.  Would this be correct? Am I missing anything? Show the existence of a certain subgroup of F2 Very similar to , except for Java.  What is the recommended way of encoding strings for an XML output in Java. The strings might contain characters like "&amp;", "&lt;", etc. What does it mean when it says under a journal article that it has been communicated by "XYZ" where XYZ is not the author but some other scholar with a very strong reputation? What is the relationship to the actual author and/or the content? Is this some sort of seal of approval to get results out and known quickly? (I am specifically wondering in the context of mathematics and mathematical physics.) Show that there is $\xi$ s.t. $f(\xi)=f\left(\xi+\frac{1}{n}\right)$ Let $f:[0,1]\to\mathbb{R}$ be continuous and $f(0)=f(1)$. Show that for any integer $n\geqslant 2$, there is $\xi\in(0,1)$ s.t.$f(\xi)=f\left(\xi+\frac{1}{n}\right).$  I think this requires the intermediate value theorem somewhere. Do particles in a sound wave ever move transversally? I'm trying to visualize how sound waves work and I was curious about something.   So sound moves in longitudinal waves, which I think I understand.  There is a really good  explaining this.  In the video the air particles are depicted as moving from left to right (longitudinally). But do they ever move up and down even a little bit? Are they on some kind of fixed axis where they can only move longitudinally and never move even the slightest bit in a vertical direction? Why is the potential energy equal to the negative integral of a force? Why is the potential energy equals to the negative integral of a force? I am really confused with this negative sign. For example, why there is a negative sign in the gravitational potential energy and what does it mean?   I read that the negative sign means you do the same force but in the opposite direction.  Doesn't that mean the object shouldn't move? Is there a free site to play Yu-Gi-Oh? I have a Canon 7D, which has a crop factor of 1.6x.  When using a Canon EF-S lens I get the focal length stated on the lens, when using a Canon EF lens I get the focal length times 1.6 when used on the APS-C body. At least that's what I thought.  I just compared two lenses:   Canon EF-S 15-85mm f3.5-5.6 Canon EF 70-200mm f4   .  The top photo is shot with the 15-85, the bottom photo with the 70-200 - both lenses are set to 70mm. Now why do they look the same?  I thought I would get 112-320mm (70mm*1.6 crop factor). What did I misunderstand about the crop factor calculation? Lightroom also displays the photos as having "70mm" and not 112mm. Is $\sum_{n=1}^\infty \frac{\sin(2n)}{1+\cos^4(n)}$ convergent? As I was just checking this 'child prodigy' out on Youtube, I stumbled upon  video, in which Glenn Beck asks the kid to do the following proof:    Further on, the kid starts sketching a proof (without a shadow of a doubt regarding the accuracy of his solution ) including the Integral Test. I don't know much about improper integrals since I just finished highschool, but this integral approach seemed, intuitively, pretty inaccurate to me since this is not a strictly decreasing function. Then I found this out from Wikipedia !   Furthermore, the blunt assessment that the series are convergent seems dubious as well..Upon a few computations of my own(mostly partial sums) , I tend to believe that the series are, in fact, divergent .  Can anyone suggest a rigurous take on this problem (easy as it may seem to some amongst you) ? How do I install different (upgrade or downgrade) PHP version in still supported Ubuntu release? I am trying to update to install PHP 5.5, 5.6 or 7.0 in Ubuntu 12.04/14.04/16.04 LTS, and I can only get PHP 5.3.10 for Ubuntu 12.04, PHP 5.5.9 for Ubuntu 14.04 or PHP 7.0.4 for Ubuntu 16.04 using official repositories (using apt-get).  I'm not really sure how to do a manual update – as I need:   to play around with the new(est) PHP features to install older PHP version due support in the software What causes a java.lang.ArrayIndexOutOfBoundsException and how do I prevent it? How can I automatically center an image? I use:  \documentclss{mwrep} ... \begin{figure}   \includegraphics{a.jpg} \caption[Long caption]{Caption} \label{pic-a} \end{figure}   How can I make images to be always visible on the center of page? (To be more specific: the center of the image should be between the left and right margin - I use different left and right margins). I don't want to put anything after every \begin{figure} in my document.  Important: I don't want to waste an entire page on a single image at the very center of the page. I want only horizontal alignment. (left &lt;-> right) How to implement Enums in Ruby? What's the best way to implement the enum idiom in Ruby? I'm looking for something which I can use (almost) like the Java/C# enums.