Show that if $p$ and $q$ are primes $\equiv 3$(mod $4$) then at least one of the equations $px^{2}-qy^{2} = \pm 1$ is soluble in integers $x, y$.
How to deal with a Pell's equation type problem with two primes
Legality of using names in a game? What font is used in this image? Why is "glide symmetry" its own type? Artin's Algebra pages 155 &amp; 156 list the types of symmetry of a plane figure as:   Reflective Rotational Translational Glide   He then goes on to say "Figures such as wallpaper patterns may have two talk about other figures having combinations of independent" symmetries. EDIT: He says "... having combinations of independent translational symmetries". See Joriki's answer.  Why doesn't glide symmetry count as having two independent (reflection + translation) symmetries? If we're going to count combinations, why not have rotation + translation etc. as their own types too? I think the questions says it all! :)  By TeX, I mean any derivative of TeX as long as the code is clearly a derivative of TeX, i.e. that it could belong on TeX.SX (and not pure Lua for example).  I'll give two answers myself so you see what I mean. What allows Star Wars ships to slow down? In the universe of Star Wars, many ships (excluding the X-Wing - the large circles on the front are retro thrust nozzles) appear to lack forward-facing thrusters or retro thrusters.    The TIE  fighter has 2 engines, . So, what slows it down?  I read on Wookieepedia that they turn using etheric rudders, and read on a chat forum (I do not remember which) that they can use etheric friction to slow down, but I couldn't find any references to this. Do they perhaps use special repulsors to slow down? In  many people answered that they use special repulsors to control their flight and direction, so perhaps could the same technology be used to slow them down?  So, how do Star Wars ships slow down?  To clarify, I want answers based on the Star Wars universe, not real-life physics. I am also willing to accept answers based on the Star Wars Extended Universe (now named Star Wars: Legends). Do sets, whose power sets have the same cardinality, have the same cardinality? Is it generally true that if $|P(A)|=|P(B)|$ then $|A|=|B|$?  Why? Thanks. An instructor has an exam of 5 questions and a question bank of 50 questions. Are there $50\choose 5$ or $(50)_5$ possible exams? Why is Android refusing to resolve DNS records pointing to internal IP addresses? Apparently this is a  and should hopefully be fixed in an upcoming update.    I have installed SQL Server Management Studio v17.9 on a clean Windows 10 installation.  This associated all .sql files with SSMS.  I want to associate .sql files with a plain text editor (eg: Sublime) but I am unable to change the default application.  I have tried:   Right click .sql file -> Open With -> Sublime + "Always use this program". Settings -> Default Applications by File Type -> Find .sql -> Change to Sublime Right click .sql file -> Properties -> "Change" button -> Sublime. Using ftype via elevated command prompt as seen    In all 3 cases, my selection is not reflected in any way, anywhere.  The icon remains SSMS, and the file opens with SSMS.  Is there some kind of "File Association retention" setting within SSMS I am unaware of?  Edit: I can change the default application for non-SSMS-associated extensions without issue.  Edit: after deleting registry key HKCU\Software\Microsoft\Windows\CurrentVersion\Explorer\FileExts\.sql, .sql files show a VSCode icon:    If I then double-click New Users.sql, it opens with SSMS and a refresh of regedit shows the .sql entry recreated with ssms.exe associated (the icon remains VSCode):    If I then right-click -> open with -> Choose Another App -> Select Sublime and hit "Always use this to open .sql files", it opens with Sublime and a refresh of regedit shows a new "b" entry created for Sublime.  The icon remains VSCode, and double-clicking opens it in SSMS:    If I then delete the a entry for ssms.exe and rename b -> a such that only sublime exists, then double-click the .sql file, it opens with SSMS and generates a new b entry for itself: The security guard problem There was a security guard in a bank. In front of him were 100 lockers in rows of 10. He thought of something as he saw all the lockers were closed. He started opening all the lockers whose lock numbers were multiples of 1, then closed all lockers whose number was a multiple of 2,and did the same for 3 and so on. If the locker was open, he closed it, if closed, opened it. He went to open multiples of 1 (all),then 2,4,6,8...and then 3,6,9.. And then finally reached 100.  After all the opening and closing,in the end, how many lockers are open and how many closed? Create Case thread Id using Apex (SOQL limit-proof) There are many posts about creating Case threadIds with Apex, notably   As the posters have noted, the SFDC rules of the threadId are subject to change over time and the only reliable way that I've seen is to do   // To be replaced with a real case Id. Id caseId = '500c0000008kJL5';   List&lt;Messaging.RenderEmailTemplateBodyResult&gt; renderResults =   Messaging.renderEmailTemplate(null, caseId, new List&lt;String&gt;{'{!Case.Thread_Id}'});  System.debug('this is the Case Thread Id: ' + renderResults[0].getMergedBody());   HOWEVER ... The method call Messaging.renderEmailTemplate burns a SOQL ()  So you can't prepopulate your Cases in an afterInsert or beforeUpdate Trigger because a transaction can be up to 2,000 (Platform Events), up to 10,000 (direct Apex DML) or more, typically, 200 rows.  All of these are > 100 SOQL limit.     So, before I go to the trouble of doing this in a chained set of async   tasks; is there more recent thinking on this subject of creating ThreadIds? Is it safe to replace a 4400mAh laptop battery with a 8800mAh battery? I have an Acer Aspire 5536G. My battery is having problems so I am thinking of replacing it. I have searched online through many sites and I found some good deals. However, currently my laptop has a 4400mAh lithium ion battery so I was thinking of replacing it with 8800mAh 12-cell battery that lasts more hours than 4400mAh without power plug. Can we interchange the sentences " What's the date today?" and "what day is it today?"  In grammar is it correct to use these sentences interchangeably?  For example "What's the day today?" and "what date is it today?" Why is $\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}$? How to center text vertically within a textbox in Photoshop It's pretty annoying that you can't center text vertically in Ps automatically, like you can in Indd. It's even more annoying that the bounding box of text can't be 'collapsed' or 'fit to text', like in Indd.  I could convert the text to shape, but then I lose coloring. I could convert to smart object, but then its not vector anymore, for when I export to pdf.  Is there a work around anyone knows of that would make it easier to center text vertically? Proof that in a field $x=0$ is equivalent to $x=-x$ Complex analysis : If $z =re^{i\theta}$, then prove that $|e^{iz}| =e^{-r\sin\theta}$ Problem :   If $z =re^{i\theta}$, then prove that $|e^{iz} | =e^{-r\sin\theta}$   My working :   $z = re^{i\theta} = r(\cos\theta + i\sin\theta)$   $\Rightarrow iz = ir(\cos\theta +\sin\theta) $ = $-r\sin\theta +ir\cos\theta $  $\Rightarrow e^{iz} =e^{(-r\sin\theta +ir\cos\theta)} = e^{-\sin\theta} e^{ir\cos\theta} $   $\Rightarrow |e^{iz}| = |e^{-r\sin\theta}||e^{ri\cos\theta}|$   Now please guide how to proceed further to get the result... thanks. Let $f$ be a holomorphic function defined on $\Bbb C$ such that $f^{-1}(\Bbb R)=\Bbb R$. Prove that there exists $a,b \in \Bbb R, \; a \neq 0 \;$ such that $f(z)=az+b \;$ for every $z \in \Bbb C$.  I was given a hint: define the function $g(z)=\frac{f(z)-f(0)}{z}$. such $g$ is also holomorphic in $\Bbb C$ since the only singularity is in $0$, and it is a removable singularity since we can define $g(0)=f'(0)$. Next I thought about proving $g$ is bounded, and thus constant (using Liouville's theorem), but pretty much out of ideas. How can I finish the proof/solve in another way? How to get more complete references Sometimes when I work I write something like  \begin{proposition} \label{my new result} ... \end{proposition} ... blah blah in Proposition \ref{my new result} blah...   but then I realize that the Proposition should actually become a Lemma. So I change it to  \begin{lemma} \label{my new result} ... \end{lemma}   The problem is that I then have to hunt down all the places where I wrote Proposition and change it to Lemma, which is tedious and error-prone. Well, we have search facilities in editors, so it is not really that complicated, but still it feels the wrong way to do this. I would be happier to write  \begin{proposition} \label{my new result} ... \end{proposition} ... blah blah in \labelNameFor{my new result} \ref{my new result} blah...   where labelNameFor is a hypothetical magic command which outputs "Proposition". Then I could forget about all the places where I reference the result and just make any changes I wish.  Is there a way to do something like this? As far as I understand the label/ref mechanism is simply based on counters, so I do not see a straightforward way, but maybe someone more clever and industrious then I am has given the problem some thought. Use a rectangular contour to evaluate the integral $$\int_{-\infty}^{\infty} \frac{\cos(mx) dx}{e^{-x}+e^x} = \frac{\pi}{e^{m\pi /2}+e^{-m\pi /2}}$$  I need to evaluate the above integral specifically using a rectangluar contour and at some point applying the residue theorem. I should be ok with the residue part, but I'm confused at to how to get this into the example form shown in the book:  $$I =\int_{-\infty}^{\infty} \frac{e^{ax}dx}{1+e^x}=\frac{\pi}{\sin(a\pi)} $$  I tried using the identity $\cos (mx) = \frac{e^{ix}+e^{-ix}}{2}$ but I end up with $\frac{1}{2} \int_{-\infty}^{\infty}\frac{e^{ix}(e^{imx}+e^{-imx})dx}{1+e^{2ix}}$, which almost resembles the example but not quite...