Let $a,b,c \in \mathbb{Z}$.  Proove that if $a^2 - b^2 = c$ then exists $m,n \in \mathbb{Z}$ which are both even/odd such that  $a = \frac{m+n}{2}, b = \frac{m-n}{2}, c = mn$  I think I should use Fermat's theorem, but I'm not sure how to do it
I have been working on the following problem:     For a given $c\in \Bbb Z^+$, find $a,b \in \Bbb Z^+: c=a^2-b^2$.   I have already figured out and proved a number of things:   A way to directly determine if a particular instance is solvable:  $\exists a,b \in \Bbb Z^+: c=a^2-b^2 \iff c \not\equiv 2 \mod 4$ An upper bound on the values for a (and b): $b \lt a\le\lfloor\frac{c-1}{2}\rfloor$ And a simple algorithm to find a solution:     $$ \begin{align*}  (a_0, b_0) &amp;= (0, 0) \\  (a_{n+1}, b_{n+1}) &amp;= \begin{cases}   (a_n+1, &amp;b_n&amp;)&amp;\text{if } a_n^2-b_n^2 &lt; c\\   (a_n, &amp;b_n+1&amp;)&amp;\text{if } a_n^2-b_n^2 &gt; c\\  \end{cases} \end{align*} $$   This can be iterated until a solution is found or the upper bound is reached.   I am currently trying to work out a more direct way to find an answer, but I have been unable to figure out any other ways to approach the problem.  How can I more directly find values for $a$ and $b$ for a given value of $c$?
Reasons why not to update an Ubuntu version to a newer/older kernel version Towards the end of , how do I get  $$ \frac{\sqrt{8z + 1} - 1}{2} &lt; w + 1 $$  Here $w = x + y \geq 0, t = (w^2 + w)/2 , z = t + w, x,y \in \mathbb N_{\geq 0}$.  Thanks. I have a 2&nbsp;GB Kingston  card which is about three years old. I put it in a reader today in my Windows Vista computer, wrote a 32&nbsp;MB file onto it, safely removed it, and then tried to read it elsewhere. Nothing. Putting it back in the Windows Vista computer it now says:     You need to format the disk in drive F: before you can use it.   What should I do? I have access to many computers and OSes if your recommendations need that. I would be very sad if I lost all the contents of the card. Most of the data is backed up, but there are a few things that aren't. :(  Doing a   # dd if=/dev/sdg of=~/tmp/sd.bin   gives me a 2&nbsp;GB file, and  the file it seems like lots of my data is still there. How can I put it back together? Remove paint from the head tube I'm planning to build a new bike from scratch.  Part of that is buying a new frame, and possibly facing/reaming it. That's actually the part I'm scared of the most.  Fortunately, the brand I'm targeting, Surly, seems to do that in the factory. But as they state, there may be some paint left where it should not be:     Since Surly frames are faced prior to painting, it is possible to   shave the paint off the head tube and BB shell ends with a carpet   knife blade.  This usually results in an acceptable result, saving you   money, but we still recommend having it done the proper way to ensure   longest life from your components.   So what is the "proper way" to remove the extra paint, if the frame is properly reamed/faced already?  Should I choose not to do it the proper way, which components are going to suffer? I suppose at least the headset and bottom bracket, but could the frame suffer as well? Example of $f,g: [0,1]\to[0,1]$ and Riemann-integrable, but $g\circ f$ is not? We can apply l’Hôpital’s Rule to the indeterminate quotients $ \dfrac{0}{0} $ and $ \dfrac{\infty}{\infty} $, but why can’t we directly apply it to the indeterminate difference $ \infty - \infty $ or to the indeterminate product $ 0 \cdot \infty $?  Furthermore, why can’t we call $ \infty + \infty $ and $ \infty \cdot \infty $ indeterminate forms?  I’m new to calculus, so please clear up my concepts if you can. Thanks! pgfplots: lines and markers in different colors With pgfplots, I am creating a simple coordinate plot with error bars like this:    \documentclass[crop,10pt]{standalone} \usepackage[english]{babel} \usepackage{pgfplots}      \begin{document}%         \setlength{\linewidth}{246pt}%              \begin{tikzpicture}             \begin{axis}[             width=\linewidth,             ytick pos = left,             xtick pos = left,             clip=false,             axis on top,             tick align = outside,             yticklabel style={%             /pgf/number format/fixed,             /pgf/number format/precision=1,             /pgf/number format/fixed zerofill         },             ytick scale label code/.code={},             xlabel={X},             ylabel={Y},     ]             \addplot[mark=*]             plot[             error bars/.cd,             y dir=both,             y fixed=0.0001     ]             coordinates{(1,0.0005) (1.1,0.0013) (5,0.0015)};             \end{axis}             \end{tikzpicture}     \end{document}   In the original plot I have a lot more points. I would now like the line and the markers (including the error bars) to have different colors to better distinguish the line from the error bars. Is that somehow possible? The only way I found so far is to plot the data just as a line in one color and then plot it again just as markers in another color. This works, but I guess there must be some better solution to this. Thanks for the support, John If I have random variables $X_1,X_2,\ldots,X_n$ that are Poisson distributed with parameters $\lambda_1, \lambda_2,\ldots, \lambda_n$, what is the distribution of $Y=\left\lfloor\frac{\sum_{i=1}^n X_i}{n}\right\rfloor$ (i.e. the integer floor of the average)?  A sum of Poissons is also Poisson, but I am not confident enough in statistics to determine if it is the same for the case above. I plan on buying Windows 7 Ultimate Edition to run on an old Pentium 4. I also want to run Windows 7 virtual machines inside Windows 7. My question is two fold:   Do I need a license for each Virtual Machine? Does the answer change if I use a third party virtualization platform (e.g. , , , etc) as opposed to ? Do SAS expanders work transparently with SAS controllers? I'm new to SAS expanders and I'm trying to get things right. I understand that SAS expanders, when used with SAS controllers which supports expanders, allows you to utilise the full capacity of supported drives on a controller.  However, say I've got a SAS controller with 4 x SSF-8087 ports, which by itself would support 16 disks. I could then, as far as I understand, instead connect each of these ports to an expander to increase disk capacity.  However, will the SAS controller "see" all disks as if they were connected directly to the controller and not through an expander. Could I look at controllers as switches in a network? Would I be able to create a RAID from a disk on expander 1 and expander 3?  UPDATE:  I don't really have a specific project where I need to use SAS expanders. I'm just curious about how they work, and was thinking about how it would work if I used one or more SAS expanders with a LSI MegaRAID controller (9260-16i) which is running in one of our servers. However, I haven't really put any thought into the distinction between "local" disks and disks in another enclosure. The LSI MegaRAID controller has 4 SFF-8087 ports, which makes it capable of accomodating 16 drives. Combinatorial Proof Question: Vandermonde convolution identity I'm really iffy on combinatorial proofs in general and now that there is a sum, it's just confused me even more. Can someone try and walk me through this proof?  $$ \binom{m + n}{r} = \sum_{k=0}^r \binom{m}{k}\binom{n}{r - k} $$ Appropriate data transformation Does $A$ a UFD imply that  $A[T]$ is also a UFD? I'm trying to prove that $A$ a UFD implies that $A[T]$ is a UFD.   The only thing I am sure I could try to use is Gauss's lemma.  Also, how can we deduce that the polynomial rings $\mathbb{Z}[x_1,\ldots,x_n]$ and $k[x_1,\ldots,x_n]$ are UFDs? I don't know what exactly it's called, by cache refresh I mean, refresh the page after clearing its cache. I don't want to clear the entire browser cache.  I can't seem to cache refresh my pages. In Firefox, I know it to be Shift+Refresh.  In Chrome, I've tried Ctrl+R, Ctrl+Refresh, Alt+Refresh, Shift+Refresh but none of them work. How to make terminal autocomplete when there are several files/directory? If I have several directories, like:  afoo abar   sometimes my terminal will refuse autocomplete when I press tab (e.g. "cd a" then tab), and print the list of directories instead. Sometimes it even throws a noisy, annoying sound. Any idea how to make it autocomplete in cases like this? E.g it can show abar first, and then afoo if I press tab again. I saw this is the case in windows, or some applciation in Ubuntu What is a good use case for static import of methods? Just got a review comment that my static import of the method was not a good idea. The static import was of a method from a DA class, which has mostly static methods. So in middle of the business logic I had a da activity that apparently seemed to belong to the current class:  import static some.package.DA.*; class BusinessObject {   void someMethod() {     ....     save(this);   } }    The reviewer was not keen that I change the code and I didn't but I do kind of agree with him. One reason given for not static-importing was it was confusing where the method was defined, it wasn't in the current class and not in any superclass so it too some time to identify its definition (the web based review system does not have clickable links like IDE :-) I don't really think this matters, static-imports are still quite new and soon we will all get used to locating them.  But the other reason, the one I agree with, is that an unqualified method call seems to belong to current object and should not jump contexts. But if it really did belong, it would make sense to extend that super class.  So, when does it make sense to static import methods? When have you done it? Did/do you like the way the unqualified calls look?  EDIT: The popular opinion seems to be that static-import methods if nobody is going to confuse them as methods of the current class. For example methods from java.lang.Math and java.awt.Color. But if abs and getAlpha are not ambiguous I don't see why readEmployee is. As in lot of programming choices, I think this too is a personal preference thing.   Thanks for your response guys, I am closing the question. Cleric build to maximize melee damage and spell healing Is it possible to detect, on the client side, whether the user is using an encrypted page or not?  Put another way -- I want to know if the URL of the current page starts with http or https. Delete commits from a branch in Git "Open With" dialog ignores my selection I am running Windows 7 x64. I have a file type that I want to associate with Microsoft Access 2003.  I right-click on the file and select Open With->Choose Default program. In the "Open With" dialog, I select the browse button. I choose Program Files (x86)\Microsoft Office\Office11\msaccess.exe and press "Open".  I end up back at the "Open With" dialog, but nothing has changed. Access is not listed as a program to use to open the file with. The original program (Adobe Acrobat) that is associated with this program is still selected.  Any tips on getting this association to work?