Question: What are the integer solutions to$$x^2+7=y^3\tag1$$   Through Wolfram Alpha, there seems to be only two solutions. Namely, $$\begin{align*} &amp; (x,y)=(1,2)\\ &amp; (x,y)=(181,32)\end{align*}\tag2$$ So I'm wondering about how would you find those solutions. And is there a way to use some sort of transformation to make $(1)$ into a more recognizable form.  And furthermore, is there a formula to determine other type solutions similar to what you already have?
The equation $y^2 = x^3 + k$ for $k = (4n-1)^3 - 4m^2$, with $m, n \in \mathbb{N}$ and no prime number that p is congruent to 1 modulo 4 divids m, doesn't have any answer and its proof can be obtained by using quadratic reciprocity law.  Do you know answers of this equation for two or three different values of $k$? In addition, do you know any reference about that?
One of the assumptions I make as GM is that the players should always be in the spotlight.  Sometimes the players find themselves in a situation where they watch NPCs speak to each other.  I don't like using this kind of situation, because it forces the players to watch me talk to myself and that turns them from active participants into an audience.  Usually I handle it by avoiding the situation.  I try not to write scenes that the players watch.  Instead the players speak to the NPCs one at a time.  When I do come across an exchange between NPCs I opt to summarize the conversation instead of acting it out.    I don't like this limitation though.  It gets a little contrived when all conversations in the game feature 5 PCs and 1 NPC.  Any suggestions? Either how to avoid doing it, or how to effectively role play a conversation between two (or more) NPCs that the characters may or may not be interacting with?   How do you keep clear which character is speaking? How do you make sure that the conversation feels like a real one and not like one person playing both roles? How to render an ASP.NET MVC view as a string? I want to output two different views (one as a string that will be sent as an email), and the other the page displayed to a user.  Is this possible in ASP.NET MVC beta?  I've tried multiple examples:  1.      If I use this example, I receive the "Cannot redirect after HTTP   headers have been sent.".   2.      If I use this, I seem to be unable to do a redirectToAction, as it   tries to render a view that may not exist. If I do return the view, it   is completely messed up and doesn't look right at all.   Does anyone have any ideas/solutions to these issues i have, or have any suggestions for better ones?  Many thanks!  Below is an example. What I'm trying to do is create the GetViewForEmail method:  public ActionResult OrderResult(string ref) {     //Get the order     Order order = OrderService.GetOrder(ref);      //The email helper would do the meat and veg by getting the view as a string     //Pass the control name (OrderResultEmail) and the model (order)     string emailView = GetViewForEmail("OrderResultEmail", order);      //Email the order out     EmailHelper(order, emailView);     return View("OrderResult", order); }   Accepted answer from Tim Scott (changed and formatted a little by me):  public virtual string RenderViewToString(     ControllerContext controllerContext,     string viewPath,     string masterPath,     ViewDataDictionary viewData,     TempDataDictionary tempData) {     Stream filter = null;     ViewPage viewPage = new ViewPage();      //Right, create our view     viewPage.ViewContext = new ViewContext(controllerContext, new WebFormView(viewPath, masterPath), viewData, tempData);      //Get the response context, flush it and get the response filter.     var response = viewPage.ViewContext.HttpContext.Response;     response.Flush();     var oldFilter = response.Filter;      try     {         //Put a new filter into the response         filter = new MemoryStream();         response.Filter = filter;          //Now render the view into the memorystream and flush the response         viewPage.ViewContext.View.Render(viewPage.ViewContext, viewPage.ViewContext.HttpContext.Response.Output);         response.Flush();          //Now read the rendered view.         filter.Position = 0;         var reader = new StreamReader(filter, response.ContentEncoding);         return reader.ReadToEnd();     }     finally     {         //Clean up.         if (filter != null)         {             filter.Dispose();         }          //Now replace the response filter         response.Filter = oldFilter;     } }   Example usage  Assuming a call from the controller to get the order confirmation email, passing the Site.Master location.  string myString = RenderViewToString(this.ControllerContext, "~/Views/Order/OrderResultEmail.aspx", "~/Views/Shared/Site.Master", this.ViewData, this.TempData); How can I disable a tab inside a TabControl? Is there a way to disable a tab in a ? Evaluate: $\lim_{n\to\infty}\left({2\sqrt n}-\sum_{k=1}^n\frac1{\sqrt k}\right)$ Stone-Weierstrass theorem with $p(1/x)$ 90s (or early 2000s) scifi (mini)series where a boy and old man chase a black guy in metal suit I'm looking for a 90s (or early 2000s) scifi (mini)series where a boy and old man chase a black guy in metal suit. I think they were going through different dimensions. I remember part of it, but I don't think I ever saw how it ended. I remember my friend telling me there was a twist at the end. I think they were trying to find something and he was always ahead of them.. I can't read beta sites' chat logos! Here, you try:  If it weren't for the text shadow, it might as well be entirely invisible...  The cause is a CSS rule lurking in the transcripts for beta sites only:  #transcript-logo {   background: none repeat scroll 0 0 #273A5D; }   The problem is the #273A5D - a bluish colour not dissimilar to that used for the site name in the beta theme. Disabling that rule yields the much more manageable  I'll admit it is a little hard to read the "beta" in that shot. But if you had to choose which part was readable, wouldn't you choose the bit that tells you the name of the site?  Edit: It happens when actually in a chat room too! Electric Potential produced by Hollow Sphere Let there be a hollow sphere (Throughout the question we will ignore the thickness of the sphere) which is positively uniformly charged,q of radius,  R. Suppose there to be a test positive charge, q' at point P which is r distance away from the hollow sphere(r>R, P is outside of hollow sphere).  Now I have an doubt on evaluating Electric Potential,V at point P.  Both source charge,q and test charge,q' are +ve, so source charge will repel the test charge which will make the charge, q' to go to infinity.I have to apply external force, F' to displace the charge, q' from infinity to point P against the equal but repulsive force, F.In short, F' is  displacing the q' from infinity to P.  So External Force, F' will do +ve work and Repulsive Force, F will do -ve work. This way the Electric Field,E produced by hollow sphere is Opposite to Displacement of test charge(i.e.,θ=180).  Now we know that E=k(q/(r^2))  [Please Notice that infinity is the lower limit of integration]  Also V=-∫r∞(E dr cosθ)=-∫r∞(E dr cos 180)=∫r∞(E dr)  At last I ended up to V=- k(q/r). I know that V=k(q/r).  I just want to know why I am having this shitty negative sign in my equation. Do I have to leave the Schengen area if I have a short term (C) visa that will expire right before my National Visa (D) officially starts? Searching for a string on multiple zip files Detecting pattern at the end of a line with grep Remove note about sudo that appears when opening the terminal Whenever I open the terminal,   To run a command as administrator (user "root"), use "sudo &lt;command&gt;".   See "man sudo_root" for details.   appears before the terminal. How do I get that to go away? Why use foreign keys if cascade action is not need? Print file content without the first and last lines Is there a simple way I can echo a file, skipping the first and last lines? I was looking at piping from head into tail, but for those it seems like I would have to know the total lines from the outset. I was also looking at split, but I don't see a way to do it with that either. How do I hide posts matching my ignored tags? Where and how do I go about hiding the posts that match any of the tags on my ignored list? The direction of static friction? In my book, the following are described to be the steps that one must follow if he wishes to find the direction of static friction force acting on an object:  &quot;(1) Draw the F.B.D(free body diagram) of the object in question with respect to the other object on which it is kept. (2) Include the pseudo force also, if the contact surface is accelerating. (3) Decide the direction of the resultant force and resolve this force into two components; one along the surface of contact and the other along the normal to this surface. (4) The direction of static friction is opposite to the component of the result force along the contact surface.&quot;  Firstly, are these steps correct? Will I always obtain the correct answer by following these steps? Secondly, consider the following problem:  In questions like these, when I try to find the resultant force and then assign the direction to static friction, I find that my answer is incorrect! Are there some other steps to follow when I know that the object is definitely accelerating in one direction or another? Edit 1: I approached the above question by doing exactly what my book suggests; first, I identified all the forces acting on the smaller block, being the normal force applied by the block A and its weight mg downward. Adding these two forces, I get another force that actually acts on the body like so:  I wonder if I'm not supposed to take either of the contact forces into consideration when drawing the F.B.D to figure out the direction of static friction acting on the body. Also, are there other steps for finding the direction of kinetic friction? Edit 2: From Judge's answer, I have been able to figure out my mistake, it being that I completely missed step (3) and didn't resolve the resultant force into components! Sheesh. Sorry! Much thanks in advance! :) Regards. Can I sync or transfer save games and settings from one platform to another? I would like to know if the same Minecraft account used on a computer can be used on a different platform like the Xbox 360? I am not asking about playing on each simultaneously as each platform has been purchased - I am asking if you can cross play on multiple platforms with the same account. Basically I would like to know if all the saved stuff and settings from one platform will carry over to another. How do I give a villager trades with custom NBT tags Proving Geometric Hahn-Banach Implies Analytic Hahn-Banach I've come across a proof online of the H.B theorem using the goemetric version.   There is a step in the proof which im not sure why is true and it is as follows:   Let $X$ be a linear space and $M \subset X \times \Bbb R$ be a maximal subspace.  Then $M = G(F)$ for $F :X \to \Bbb R$ linear.   Someone can show me why this is true?   Maybe I misunderstood the proof, its the last paragraph of Matrin's answer here -.   Thanks a lot for helping! What are some typical LuaTeX gotchas?