let us suppose that there is given  such  kind of conjecture   if  their division is  square of integer  then we  will have  $\frac{a^2+b^2}{ab+1}=k^2$  for some  integer $k$, then  we would have  $a^2+b^2=k^2*ab+k^2$  or  we would have  $a^2-k^2*a*b+b^2=k^2$  compare  to the equation  $a^2-2*a*b+b^2=(a-b)^2$  we would have  $k^2=2$     but is that proof ? i could   not understand  if there is  another option  or alternative ways
Let $a,b$ be positive integers.  When $$k = \frac{a^2 + b^2}{ab+1}$$ is an integer, it is a square.    Proof 1: (Ngô Bảo Châu): Rearrange to get $a^2-akb+b^2-k=0$, as a quadratic in $a$ this has two values: $a$ and $kb - a = (b^2-k)/a$. (The second root is determined in two different ways from the expansion $(x-r_1)(x-r_2) = x^2 - (r_1 + r_2)x + r_1 r_2$.)  Now suppose we have $a,b$ such that $k$ is an integer but not a square, by the investigation about roots we have that the second root is a nonzero integer since $k,b,a$ are integers and $k \not = b^2$, futhermore it is positive which is easily seen from its defining equation.  WLOG assume $a \ge b$ so that the second root is strictly smaller than $a$. This leads to a decent, replacing $a$ with the second root.    Proof 2 (Don Zagier): Apply reduction theory (specifically, Sätze 1 and 2 of Section 13 of my book on quadratic fields) to the quadratic form $x^2 + kxy + y^2$, which is the unique reduced quadratic form in its equivalence class.    Note that Proof 2 is pretty much the same as Proof 1 when written out in explicit detail, but I could not read Zagier's book because I cannot read German.  I would like to know more approaches to this and other alternative proofs of this result if possible! Thanks in advance.  I would also be interested in related problems (especially easier ones of a similar nature) and texts which cover the reduction theory in English.
Nautilus context menu for current directory when there's lots of files / "Open in Terminal" I asked a question almost as soon as I got my account. I got a few decent answers, but none of them helped solve my problem. The question has long since become buried by newer questions, and there's still no accepted answer. Unfortunately, because there are quite a few answers with votes, to a casual observer it probably looks like the question was answered and I just neglected to mark an answer as accepted, and the system does not consider it "unanswered".  Or, I asked a question that was a bit more difficult to answer. After waiting for some time, I received no answers, little or no comments, and only a few views. The question does show up in the "unanswered" list, but no one has answered my question.  In either case, what's the protocol for me to try to get this question answered?  I see several options:   Post the question a second time, possibly linking to the original, to get attention for the original question. This is encouraged on sites like Reddit.  However, it's probably not appropriate here, since we're encouraged to check for duplicates and not submit a question that's already been asked by someone else, so this probably falls under that heading. Wait for some kind of wiki-like talk page to be implemented, then post a plea on the talk page for input and hope that activity on the talk page bumps up the question. Edit the question, but do nothing further on the Stack Exchange site. If I edit my post to say that I haven't accepted any of the current answers but am looking for more, then the question might get answered eventually by people browsing through old questions to find something they can answer. Post the question on other sites, but link to my original question on Stack Exchange. People will either answer here, or they'll answer on the other site and I can post the accepted answer here myself.  This might be considered spamming, so it might be better to do this without the link.      For more information, see the "" page in the . Capital letters in "Theorem", "Conjecture" etc This is my first post and as you can probably guess I am a mathematician so I have no clue about grammar. I am writing a mathematical document at the moment and I would appreciate some advice on my usage of capital letters. Can somebody please confirm if the following are correct?  1) "We know by T​heorem 4.6.1 that..."   2) "the following c​onjecture, due to..."  3) In order to prove the "Mckay-Stevens $k$-covering C​onjecture, once needs to show that..."  I'm particularly confused about 3).  Many thanks for any help Is abstract algebra (mostly?) restricted to $2$-ary operators? We all know that Superman changes in the phone booth. But where does he put his regular clothes while he's on a mission? He doesn't throw it out, because he changes back. But we don't see him carrying a bag of dirty laundry while he's on a mission.  ("In his right hand, Superman has Lois Lane, and in his left hand he seems to be holding... boxers??)  And I can't imagine he leaves it in the phone booth.  So where does Superman leave his regular clothes after he changes? Variance of the future value of a cash flow I want to compute the variance of $FV =A\sum_{k=1}^n(1+r)^k=:g(r)$, assuming interest rate $r\sim\mathrm{N}(\mu,\sigma^2)$ and constant equal payments $A$. As discussed in the comments, the Delta Method gives an approximation $\mathrm{Var}[FV]≈\sigma^2\times(g′(\mu))^2$. Is there a way to get an exact answer? Adding a CMS to an existing Magento shop I am working on a project for 3 niche stores built on magento (using magento's multi-store function)  that each get roughly 50k unique visitors a day.  The sites don't currently have a blog or forum or any social networking aspects.    Would like to add a cms to each site that can be centrally run and would like it to take over the front end content from Magento.  Also would like it to maintain an online blog/publication of sorts with videos, articles, and the like with privileges to edit the content given to a dozen or so people with different privileges.  Want to add a forum to each site that is fairly robust and to possibly add some social networking aspects down the road, so extandability and available plugins/mods in each cms is important.  Other than shared login between the forums,blog/publication and store, would like to be able to integrate some content from the forums and blog/publication into the store as well.  After researching this a bit, I am inclined towards Drupal, but I haven't found any modules to integrate it with Magento.  Also, since the blog content will be done by about a dozen nontechnical people, I want something that is very easy to work with.  Lastly, since the site gets a good amount of traffic, speed and security are very important.  What CMS would you recommend integrating in this context?  Deciding between Drupal, Wordpress and ModX.  Also Plone as well.  Thanks. Force refresh view in LWC In Aura, we can do      $A.get('e.force:refreshView').fire();   to cause standard components to update.  How can we do an equivalent in LWC because sfdx refuses to publish .js code with $A in it saying the usage is disallowed in LWC?  For now i tricked it with   eval("$A.get('e.force:refreshView').fire();");   but that is obviously less than ideal.... Physics history book with some math I am looking for a book explaining physics from, say, Galileo and Newton till now; the book should be written using some math, similarly to my physics books when I was at the university (Halliday Resnik). I am searching a book that take care of explaining the true "evolution" of reasoning, discoveries, experiment to prove or disprove something.  I do not like a book that gives definitions without explaining what had happened. I also like to find, on that book, explanations on how actual measurements are taken (how can we measure the speed of light, the mass of celestial bodies, the attributes of electron, and how were the measurements taken in the past - how was possible to measure the speed of light in 1700-1800?).  I am not very interested in recent theories like strings, branes.... so it is OK if these are not discussed.  I like to find a lot of discussions on concept that are related, like for instance the implications on space dimensionalities, ether non-existence, strange facts, missing explanations. Is there a way to get the source code from an APK file? Which is correct, "neither is" or "neither are"? Bob: "Can I set the font color? Can I customize the text?"  Frank: "Neither of these options is available. Sorry!"    Is "neither is" always correct or should one use "neither are" in some cases and what are the exact rules?  of this-is-why-stackexchange-exists :P Unfortunately hard disk of my system is damaged, now I recovered my database. But it cannot be attached again to SQL management studio 2005. The error is:  myfile.mdf is not a primary database file. (SQL server , error 5171) I'm having a problem finding out a way to project from top view and the side view so I texture better. It just replaces the current projection with the top view    So is it possible to project from view twice on the same image? I would like to be able to execute a SOQL query that will allow me to see which users have access to a certain record type for a certain object.  For example, if I have a case record type called Phone Team, I would like to make a query that returns the names of all the users that have been assigned the Phone Team record type, as can be seen in the native user interface in a profile's Object Settings page.  So far I have managed to get a query that can get the names of all the users that have, for example, read access to the case object:  SELECT Assignee.Name FROM PermissionSetAssignment      WHERE PermissionSetId IN          (SELECT ParentId FROM ObjectPermissions WHERE SObjectType = 'Case'               AND PermissionsRead = true AND Parent.IsOwnedByProfile = true)   However, there doesn't seem to be any way to access record type assignment in the same way that you can object permissions.  I see from  that the info can be obtained via the metadata API, but I would much prefer to do it by SOQL if possible.  Thanks Calculate the last digit of $3^{347}$ I think i know how to solve it but is that the best way? Is there a better way (using number theory). What i do is: knowing that  1st power last digit: 3 2nd power last digit: 9 3rd power last digit: 7 4rh power last digit: 1 5th power last digit: 3    $3^{347} = 3^{5\cdot69+2} = (3^5)^{69} \cdot3^2 = 3\cdot3^2=3^3=27 $ so the result is $7$. How to convert a HEIF/HEIC image to JPEG in El Capitan? How to prevent Google from finding my admin index page? Is "connected, simply connected" Redundant? Here are my definitions of &quot;connected&quot; and &quot;simply connected.&quot;  A topological space $X$ is connected if and only if it is not the union of two nonempty disjoint open sets. A topological space $X$ is simply connected if and only if it is path-connected and has trivial fundamental group (i.e. $\pi_1(X)\simeq\{\mathrm{e}\}$ and $|\pi_0(X)|=1$).  It is a classic and elementary exercise in topology to show that, if a space is path-connected, then it is connected. Thus, if a space is simply connected, then it is connected. Yet, despite this implication, I've read several cases where the words &quot;connected, simply connected&quot; appear together. For example, in Kobayashi and Nomizu's Foundations of Differential Geometry, Volume 1, page 252, the following is written: &quot;Let $M$ be a connected, simply connected analytic manifold with an analytic linear connection.&quot; In  and  (Links to Journal Storage, also known as JSTOR), they use &quot;connected, simply connected&quot; on their first page. The first of these takes it a step further, referring to &quot;closed, compact, connected, simply connected $4$-manifolds,&quot; where a closed manifold (unfortunate terminology) is defined as a compact manifold without boundary. My question is simple: Why use the words &quot;connected, simply connected&quot; when being simply connected implies connectedness? Thank you. Is it possible to interrupt spellcasting? Is there a destructor for Java? Is there a destructor for Java? I don't seem to be able to find any documentation on this. If there isn't, how can I achieve the same effect?  To make my question more specific, I am writing an application that deals with data and the specification say that there should be a 'reset' button that brings the application back to its original just launched state. However, all data have to be 'live' unless the application is closed or reset button is pressed.  Being usually a C/C++ programmer, I thought this would be trivial to implement. (And hence I planned to implement it last.) I structured my program such that all the 'reset-able' objects would be in the same class so that I can just destroy all 'live' objects when a reset button is pressed.  I was thinking if all I did was just to dereference the data and wait for the garbage collector to collect them, wouldn't there be a memory leak if my user repeatedly entered data and pressed the reset button? I was also thinking since Java is quite mature as a language, there should be a way to prevent this from happening or gracefully tackle this.