I know that an integer is the sum of two squares if and only if it is the norm of some Gaussian integer. However, the numbers below are so large I don't even know where to begin, let alone know whether or not it can even be a sum of two squares.  5^7 * 7^9 * 11^12 * 13^11 2^10 * 11^12 * 13^15 * 23^11 3^11 * 7^10 * 13^12 * 17^13 3^8 * 7^12 * 13^5 * 19^4
I'm looking for some help disproving an answer provided on a  I posted about computing the number of double square combinations for a given integer.    The original question is from the Facebook Hacker Cup       Source: Facebook Hacker Cup   Qualification Round 2011      A double-square number is an integer $X$   which can be expressed as the sum of   two perfect squares. For example, 10   is a double-square because $10 = 3^2 + 1^2$.   Given $X$, how can we determine the   number of ways in which it can be   written as the sum of two squares? For   example, $10$ can only be written as $3^2 + 1^2$   (we don't count $1^2 + 3^2$ as being different). On the other hand,   $25$ can be written as $5^2 + 0^2$ or as   $4^2 + 3^2$.      You need to solve this problem for $0 \leq X \leq 2,147,483,647$.      Examples:      $10 \Rightarrow 1$      $25 \Rightarrow 2$      $3 \Rightarrow 0$      $0 \Rightarrow 1$      $1 \Rightarrow 1$   In response to my original question about optimizing this for F#, I got the following response which I'm unable to confirm solves the given problem correctly.       Source:       Again, the number of integer solutions   of $x^2 + y^2 = k$ is four times the   number of prime divisors of $k$ which   are equal to $1 \bmod 4$.      Knowing this, writing a program which   gives the number of solutions is easy:   compute prime numbers up to $46341$ once   and for all.      Given $k$, compute the prime divisors of   $k$ by using the above list (test up to   $\sqrt{k}$). Count the ones which are   equal to $1 \bmod 4$, and sum. Multiply   answer by $4$.   When I go through this algorithm for $25$, I get $8$ which is not correct.          For each prime factor (pf) of $25$ ($5$, $5$ are prime factors of $25$)         if pf % $4$ = $1$ (true for both $5$'s), add $1$ to count      return $4$ * count (count would be $2$ here).          So for $25$, this would be $8$
My 12.04 x64 system won't boot into GUI, but I can still use virtual console with my own account, so apparently it's the Windows Management System that's causing it, how can I reinstall it from tty, what's the apt-get command? Traditionally  at the end of a filename after uploading it via the image uploader. That doesn't seem to work any more. I've tested it on Q&amp;A as well as chat.        (Yes, I know big M doesn't work)  Could we have this fixed please? In order to protect a portal in the nether I have an obsidian room with an iron door, which works off a switch in the room. The problem is that if I leave the room and someone else on the server closes the door from the inside I can't get back to that portal.  Powering red stone on the outside of the door doesn't seem to work, as the switch inside is already powering it, so I'm guessing something like an XOR gate will be the solution, however I am finding it difficult to see how to make this work in practice. Disable drag and drop option for multiple file uploader These are the screenshots of multiple file uploader from the same SharePoint 2010 site. Some users are seeing the first and some are seeing the second one. Is it possible to disable the second view(drag and drop) by editing upload.aspx in the layouts folder?      Thank you The following question was asked on a high school test, where the students were given a few minutes per question, at most:     Given that,   $$P(x)=x^{104}+x^{93}+x^{82}+x^{71}+1$$   and,   $$Q(x)=x^4+x^3+x^2+x+1$$   what is the remainder of $P(x)$ divided by $Q(x)$?     The given answer was:     Let $Q(x)=0$. Multiplying both sides by $x-1$:   $$(x-1)(x^4+x^3+x^2+x+1)=0 \implies x^5 - 1=0 \implies x^5 = 1$$   Substituting $x^5=1$ in $P(x)$ gives $x^4+x^3+x^2+x+1$. Thus,   $$P(x)\equiv\mathbf0\pmod{Q(x)}$$     Obviously, a student is required to come up with a “trick” rather than doing brute force polynomial division. How is the student supposed to think of the suggested method? Is it obvious? How else could one approach the problem? I have got this problem on limits  $$ \lim_{x \rightarrow 0}\left(\frac{\sin(x)}{x}\right)^\frac{1}{x^2}$$ What I am doing here is that taking log and then applying L-Hospital's Rule $$y= \lim_{x \rightarrow 0}\left(\frac{\sin(x)}{x}\right)^\frac{1}{x^2} $$ then $$\ln(y)=\lim_{x \rightarrow 0} \frac{1}{x^2}\ln\left(\frac{\sin(x)}{x}\right)$$ now it becomes $\frac{0}{0}$ type , but problem is how to take derivative here? when differentiating $\ln\left(\frac{\sin(x)}{x}\right)$ it will become $\frac{x}{\sin(x)}$ then we'll have to differentiate one more time like we do in case of $f(f(x))$ ? $\sin(x)$ and $x$ will be separately differentiated or together ? Choosing equipment options at character creation So if I'm correct, when you chose your starter gear and wealth you can either choose to roll some dice depending on your class and buy everything from that or you can choose the starter gear from your class and add the background items (including some gold). Here are my 2 questions:   If I choose the gear from my class, do I have to take all the equipment from only one of the (a) or (b) choices, or can I freely mix-and-match choices from the two sets? Can you spend the extra gold you get from your background on extra equipment (e.g. some soap and some fishing equipment) or do you have to start with that gold in its pouch?   The reason I ask is that I have recently created a level 1 wood elf druid and I mixed-and-matched the gear from the class, so that I chose a shortbow as a "(b) simple weapon" for the first point and a "(a) Scimitar" at the second point. My DM wasn't so happy about this, as he can't understand why I'm mixing (a) and (b).  He also wasn't so happy with me spending my 10gp from my outlander background. I decided that my druid wouldn't lay so much worth in 10gp compared to a bunch of stuff I got for that (e.g. some soap and some fishing equipment and some other useful things) and he seems to think that I'm supposed to just start with the gold instead. mysterious error: \pdfendlink ended up in different nesting level than \pdfstartlink I ran into the "mysterious"  \pdfendlink ended up in different nesting level than \pdfstartlink   error.  According to , "this happens when hyperref is used under pdftex and a citation splits across a page boundary".  The given solution is to manual fix the citation to not split across the page boundary.  However, this workaround isn't feasible for me, as the documents are automatically generated on a server with no user interaction.  Is there any solution to the problem known, perhaps an experimental hyperref action or a patch to pdftex?  (Probably, one could use luatex to solve the problem, but for other reasons, luatex is not an option in this project.)  MWE:  \documentclass[%draft               ]{article} \usepackage{hyperref} \setlength{\textwidth}{3cm} \setlength{\textheight}{3cm}  \begin{document} Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor invidunt ut labore et dolore magna \href{http://www.tug.org/errors}{aliquyam erat, sed diam voluptua.} \end{document} I have a Python script that will need to be given to a client.   We are currently evaluating what license level the client will need in order to run the script for their software acquisition purposes.   We all have ArcGIS 10.1 Advanced license in our company.   I checked all the tools and extensions used in the script against the web help and found that they should work on the Standard license level.   Is there a way I can test the script against the Standard license level, when we have no Standard license level in or company (only the higher Advanced level)? I'm trying to setup a image decal in cycles, and I can't figure how to setup the nodes to work fine. Does anyone know the solution? How to determine when Settings change on iOS Fundamentally, what does a friction force do for a rolling object? I am very confused by the way my textbook explains it. In my textbook , it says:  A wheel rolling on a horizontal flat surface at a constant velocity experiences no friction force. Why?   On the same surface, there is an acceleration of the wheel pointing to the right (probably caused by a force), so the ball is angularly accelerating in the clockwise direction. In this case, a friction force appears, and it is also pointing to the right. How come? What does the friction do for this wheel?  On an inclined plane, a ball freely rolls down the surface. The direction of friction is up the ramp, which confuses me because in the previous example the friction force was in the direction of the wheel's acceleration.   And there is a difference if a wheel is freely rolling and if there is a torque acting on the ball's center of mass. WHY???  What does friction do in these cases? How does it cause an object to roll? When $f(x+1)-f(x)=f'(x)$, what are the solutions for $f(x)$? Why or how did Agen Kolar and Saesee Tiin seem to die so easily without any resistance? I know that there is a very similar question, but I'm not interested in highlighting. What I'm looking for is some tool in Linux to edit pdf files. I've tried pdfedit which did the work for me, but if freezes after a few edits. I've tried this tool under Ubuntu 9.10 and under KDE Fedora 14 and the same thing happens. I've tried Okular but the notations are too big to fill out exercises from pdf workbooks.    Is there any tool that suits this job? I don't care whether it is open source or I have to pay a reasonable price for it. Good reading on the Keldysh formalism I'd like some suggestions for good reading materials on the Keldysh formalism in the condensed matter physics community. I'm familiar with the imaginary time, coherent state, and path integral formalisms, but lately, I've been seeing Keldysh more and more in papers. My understanding is that it is superior to the imaginary time formalism at least in that one can evaluate non-equilibrium expectations. How do I solve this limit: $\lim _{x \to 0} \left(\frac{ \sin x}{x}\right)^{1/x}$? I use form to send POST request to a page and to login user with wp_signon() in order to authenticate user to my wordpress installation as described in WP documentation:  $creds = array(); $creds['user_login'] = $_POST["user-login"]; $creds['user_password'] = $_POST["user-password"]; $creds['remember'] = true;  $user = wp_signon( $creds, false );   After this little piece of code I'm checking if user was logged in:  if ( is_user_logged_in() ) { echo "SUCCESS"; } else { echo "FAIL!"; }   But I got FAIL! all the time. Then after sniffing around I found this little trick:  wp_set_current_user( $user ); if ( is_user_logged_in() ) { echo "SUCCESS"; } else { echo "FAIL!"; }   I've got SUCCESS on this one but when I leave this page I got FAIL again and again.  Can anyone explain me how to login user with wp_signon() without logging her out after page is changed or reloaded or whatever.    I've got desirable result when I go to /wp_admin and login with WP's default login form. I can navigate through every page of my WP site remaining logged-in all the time. But when I try to do this outside the default form I FAIL.  Guide me! PLEASE! Why AUC =1 even classifier has misclassified half of the samples? For $x,y\in\mathbb{N}$ how many ordered pairs $\left(x,y\right)$ satisfy   $\dfrac{1}{x}+\dfrac{1}{y}=\dfrac{1}{pq}$ where $p,q$ are distinct primes?