The following equation has no solution for $m&lt;1000$ (see computation below)  $$n^2 \equiv -1 \mod 4m+3 $$  Question: Is there a solution in general?      sage: for k in range(1000): ....:     for n in range(4*k+3): ....:         if (n**2+1).mod(4*k+3)==0: ....:             print([k,n]) ....: sage:
I've read through the elementary proof of why there exists a solution $x$ to $x^2\equiv -1\pmod p$ iff $p\equiv 1 \pmod 4$ for $p$ an odd prime. Is there a group theory generalization for this fact as well?
Achievements tab isn't working in Area 51 When I try to bring down the Achievements tab in Area 51, it's showing an error dialogue.  Strangely enough, the global inbox dropdown works just fine.  Could be connected to .  Screenshot: What happened to Saruman after he was knifed by Grima? How do I render a Twig template and load a CSS file from a controller? Is this considered a good practice for rendering a Twig template and loading a CSS file from a controller?   DefaultController.php  namespace Drupal\acme\Controller;  use Drupal\Core\Controller\ControllerBase; use Drupal\Core\DependencyInjection\ContainerInjectionInterface; use Symfony\Component\DependencyInjection\ContainerInterface; use Drupal\Core\Template\TwigEnvironment;  class DefaultController extends ControllerBase implements ContainerInjectionInterface  {   /**   * @var Drupal\Core\Template\TwigEnvironment   */   protected $twig;    public function __construct(TwigEnvironment $twig)   {     $this-&gt;twig = $twig;   }    public static function create(ContainerInterface $container)   {     return new static(       $container-&gt;get('twig')     );   }    /**    * hello    * @param  string $name    * @return string    */   public function hello($name)    {      $template = $this-&gt;twig-&gt;loadTemplate(       drupal_get_path('module', 'acme') . '/templates/hello.html.twig'     );      $markup = [       '#markup' =&gt; $template-&gt;render([ 'name' =&gt; $name ]),       '#attached' =&gt; [ 'css' =&gt; [           drupal_get_path('module', 'acme') . '/assets/css/acme.css'         ]       ]     ];      return drupal_render($markup);   } }   acme.routing.yml  acme_hello:   path: '/acme/hello/{name}'   defaults:     _content: '\Drupal\acme\Controller\DefaultController::hello'     _title: 'acme Title'   requirements:     _permission: 'access content'   hello.html.twig  &lt;div class="acme-hello-text"&gt;   &lt;h1&gt;Hello {{ name }}!&lt;/h1&gt; &lt;/div&gt; Whenever I open my files, my hidden files show up. I can hide them but when I open up my files again, they're showing again. How do I make the hidden files invisible permanently? Are there solutions that can limit the CPU usage of a process? I am using an old laptop and doing heavy processing that needs high CPU usage for a long time (~30mins - 2hours). When the process puts heavy load on CPU for a long time, CPU overheats and computer shuts down. I don't want to use a cooler/stand because I am carrying this laptop around and working outside home.  Is there a way/application that I can limit CPU usage of a process in Windows?  Note: Setting the priority of the process from task manager does not work.  Note 2: Fans are clean. How to find real world Selenium projects Where can I find sample or real Selenium projects. All the material I can find on github and on internet are basic, and never intermediate or advanced.  How does a real Selenium project look in real life? How can I find sample code? Im new to Illustrator and have found the clipping mask is not so easy compared to Phtoshop. Could any one tell me how i can make this line texture a clipping mask over the circle? Why does HTML require that multiple spaces show up as a single space in the browser? Xcode - How to fix 'NSUnknownKeyException', reason: … this class is not key value coding-compliant for the key X" error? How to place a .png texture with transparency over a material I am trying to create a low poly face with a face from a .png file. The issue is I already have material placed for hair and skin, as shown in the picture below.   I have created a .png file with a face which I would like to overlay over the skin material on the front of the head.    All examples I searched so far assume I also want the color of the material from the .png texture.  Update:  I added UVMapping, changed view to texture, made sure one face is on the eye but nothing is displayed..... see image... I don't know what is going on. I know that it is defined such that there is no more than one y-value output for any given x-value input, but I"m wondering WHY it is defined that way? Why can't we apply everything we know about functions to equations that have more than one y-value output for any given x-value input?   In short, what is the benefit or reason to define a function this way?  At the same time, a little historical context would also be appreciated (i.e., when the concept originated and why).  Thanks,  Moshe How can I rename many files at once? I need to rename a bunch of files in order to fix their sort order, therefore I need to be able to do a sort of "find and replace" so I can replace chunks of the filenames at a time.  What tools should I use? I prefer GUI but a command line tool recommendation would also be fine. Can I have multiple DHCP servers on one network? This is a  about Redundant DHCP Servers.   Is it possible to have more than one DHCP server on the same LAN? What are the implications of doing this?    What happens if there is more than one DHCP server available? How do my clients know which one to use?  How can I have DHCP servers supplying addresses to more than one subnet\network segment? How can I configure multiple DHCP servers to supply addresses for the same subnet. Why do we use "dried up" instead of just "dried"? The creek has dried up.   In this example sentence, you should say "dried up." If you say just "dried," it sounds incomplete. Is this an example of an idiom or is there a grammatical rule governing this instance? If so, what is it? The PHB on p. 195 (or the ) says:     When you take the Attack action and attack with a light melee weapon that you’re holding in one hand, you can use a bonus action to attack with a different light melee weapon that you’re holding in the other hand. You don’t add your ability modifier to the damage of the bonus attack, unless that modifier is negative.   The rules do not make a distinction between "main-hand" and "off-hand" weapons.  My character dual wields a shortsword with her right hand and a dagger with her left hand.  Could my character attack with her left hand dagger first, doing 1d4 + STR/DEX mod damage, then attack with her right hand shortsword, doing 1d6 damage?  Does it matter which weapon I attack with first when two-weapon fighting? In what sense are Asgardians "immortal" in the MCU? Below, the  of second-order logic is used.    My question is about a second-order analogue of $ZFC$ other than the usual "second-order $ZFC$." Rather than define the latter, I'll just state that its (set-sized) models are exactly the $V_\kappa$s with $\kappa$ strongly inaccessible (or if you prefer, the uncountable Grothendieck universes).  Specifically, I'm interested in the second-order theory consisting of the usual (first-order) axioms of Pairing, Extensionality, Union, Choice, Infinity, and Powerset, and the Separation and Replacement schemes ranging over second-order formulas. I'll call this "$ZFC_2^{scheme}$."    Specifically, for each formula $\varphi(x, y_1,...,y_k)$ of second-order logic with only the displayed variables, each of which is first-order, we have a corresponding Separation and Replacement instance. We do not allow second-order parameters here, so formulas with second-order free variables don't yield Separation/Replacement instances.   The passage from $ZFC$ to $ZFC_2^{scheme}$ has a nice uniformity: it's an example of a more general operation $\mathcal{ZFC}$ which spits out a $ZFC$-analogue given a logic at least as strong as first-order logic.    This is the reason for the parameter restriction above: I want $ZFC_2^{scheme}$ to look just like $\mathcal{ZFC}(\mathcal{L})$ for any other logic $\mathcal{L}$, and not take into account the particular feature that we now have a new kind of variable at our disposal - and in particular, take as "syntax-free" an approach to logics as we can get away with. (This is a choice of arguable value, but it's the one I'm making for now.)   It's easy to show that every model of $ZFC_2^{scheme}$ is well-founded, so to understand its models we only need to look at transitive sets. A natural hope at this point is that $ZFC_2^{scheme}$ is just second-order $ZFC$ in disguise, that is, that for a transitive set $M$ we have $M\models ZFC_2^{scheme}$ iff $M=V_\kappa$ for some strongly inaccessible $\kappa$. However, this isn't at all obvious to me (although it is easy to show the right-to-left direction).     Question. What are the (set-sized) models of $ZFC_2^{scheme}$?   (Note by contrast that the "arithmetic analogues" $PA_2^{scheme}$ and second-order $PA$ are equivalent since each pins down $\mathbb{N}$ up to isomorphism - although that leads to .)  (I'm happy to drop Choice if that would help.)  EDIT: As Hanul Jeon pointed out below,  shows that consistently $ZFC_2^{scheme}$ (there called "$ZFC_2^{def}$") has countable models, so we have a partial answer. However, I don't see at the moment a way to get an outright $ZFC$ proof that $ZFC_2^{scheme}$ is strictly weaker than second-order $ZFC$. Is there a rhetorical term for remarks like "you're exactly what I've come to expect from..."? Is there a specific term for a statement that uses words such that their precise meaning conveys a subtext contrary to the colloquial meaning?  For example, the joke in Men in Black:     Gentlemen, congratulations. You're everything we've come to expect   from years of government training.   where the precise use of the words conveys a sarcastic meaning (ie that the candidates are inept because that's what he's come to expect from government training).  I was going to use "irony" to describe this but then realized this is basically the opposite of irony - the use of words to convey their literal intention, when a nonliteral intention was expected. If I have multiple objects selected can I use scaling to make the objects closer together (without rescaling the objects)? Can I travel to the USA while working remotely for my non-US employer? I would like to travel in the USA for a period of 3 - 6 months. But I want to spend some time working for my employer (remote work) and getting paid by the employer. Is this possible under a tourist visa (B1/B2)?  I hope this is OK since I will not be getting paid by a US employer. I am in the IT trade so all that I will be needing is my own laptop and a connection to the internet. The money I earn would actually help me pay for the expenses I get during the holiday.