I need to find a good configuration for my computational kernel, which forces me to find some integer solutions to the following simple equation:  $a \cdot x - b \cdot y = c$,  where $a$, $b$ and $c$ are constant positive integers, and $x$ and $y$ are my unknown, that I need to also be positive integers.  I thought I could solve this using the Diophantine solution, but I'm not sure how to proceed due to one of the addends to be negative.  What is the best way to find all solutions to such an equation?
I want to find a set of integer solutions of Diophantine equation: $ax + by = c$, and apparently $\gcd(a,b)|c$. Then by what formula can I use to find $x$ and $y$ ?   I tried to play around with it: $x = (c - by)/a$, hence $a|(c - by)$.     $a$, $c$ and $b$ are known. So to obtain integer solution for $a$, then $c - by = ak$, and I lost from here, because $y = (c - ak)/b$. I kept repeating this routine and could not find a way to get rid of it? Any hint?  Thanks, Chan
A (highfalutin) word for a highfalutin word? To illustrate the question, badroit used the word "meretricious" meretriciously.   Is there an English word that describes highfalutin words like "meretricious" that are (gratuitously) used in place of better-understood words like "garish"?  The word may or may not be self-describing. How prove this $\sum_{k=1}^{2^{n-1}}\sigma{(2^n-2k+1)}\sigma{(2k-1)}=8^{n-1}$ Given the positive integer numbers $n$,prove that $$\sum_{k=1}^{2^{n-1}}\sigma{(2^n-2k+1)}\sigma{(2k-1)}=8^{n-1}$$  where $\sigma(n)$ is defined as $$\sigma{(N)}=\sum_{d|N}d$$  This problem I think is very interesting, and have nice methods?  Thank you everyone.  and I find some of this     or can see: Let $S$  be a square of unit area. Consider any quadrilateral whose $4$ vertices lie on each side square $S$. Let the length of the sides of this quadrilateral be $a,b,c,d$. Then prove that $$2 \leq a^2+b^2+c^2+d^2 \leq 4$$     This problem appeared in IIT JEE $1997$ (re-exam).I really do not know know how to approach this problem properly.I have managed to prove $a^2 + b^2 + c^2 +d^2  \leq 4$ but not the initial inequality. How to remove all subviews? Finding $\int x^xdx$ Word whose form is contrary to its meaning? Is there a word for a word whose form is contrary to its meaning?  For example, "quotidian" is anything but quotidian. sql ORDER BY multiple values in specific order? PDF Editing in PHP? Does anyone know of a good method for editing PDFs in PHP? Preferably open-source/zero-license cost methods. :)  I am thinking along the lines of opening a PDF file, replacing text in the PDF and then writing out the modified version of the PDF?  I have programmatically created PDF files in the past using FPDF, but found it a little unwieldy at times. Can you explain the "Axiom of choice" in simple terms? As I'm sure many of you do, I read the  webcomic regularly. The  involves a joke about the Axiom of Choice, which I didn't get.     I went to Wikipedia to see what the Axiom of Choice is, but as often happens with things like this, the Wikipedia entry is not in plain, simple, understandable language. Can someone give me a nice simple explanation of what this axiom is, and perhaps explain the XKCD joke as well? Find missing indexes tool available to find missing indexes in mssql 2008 ? can we find the missing index using Database Engine Tuning Advisor How can I remove malicious spyware, malware, adware, viruses, trojans or rootkits from my PC? What does the '!!!' syntax mean in javascript? How to read command line arguments of another process in C#? "Not a COM32R image" error when trying to install from a USB key Not a COM32R image boot   is a message that the console infinitely keeps spitting out when I try to install Ubuntu 10.04 from a USB. I created the USB using the StartupDiskCreator in 11.04.  What is the reason for this behavior? What do I do to set it right?   I have followed the steps perfectly, as enlisted in the . Why does US Customs & Immigration ask the purpose of my trip when I'm returning home? As a US citizen, when I re-enter the US, I am often asked where I was, how long I was there, and the reason for my visit.  Why am I asked these things?  Am I required to answer? Don't I have a legal right to enter the US (as long as I'm not doing or carrying anything illegal)?  If I were to refuse to answer these questions, would I be permitted entry?  Note: I'm not really bothered by the questions (except when the agent asking them is especially rude), I'm just curious. How do I correctly tag my questions? Here are some tips for tagging questions. These guidelines will help you more accurately tag your questions, which in turn will help them get more attention and lead to better answers faster.     For more information, see "" in the . If $f$ is continuous and $\,f\big(\frac{1}2(x+y)\big) \le \frac{1}{2}\big(\,f(x)+f(y)\big)$, then $f$ is convex Let $\,\,f :\mathbb R \to \mathbb R$ be a continuous function such that $$ f\Big(\dfrac{x+y}2\Big) \le \dfrac{1}{2}\big(\,f(x)+f(y)\big) ,\,\, \text{for all}\,\, x,y \in \mathbb R, $$ then  how do we prove that $f$ is convex that is $$ f\big(tx+(1-t)y\big)\le tf(x)+(1-t)f(y) , \forall x,y \in \mathbb R , t\in (0,1)? $$ I can prove it for dyadic rational's $t=\dfrac k {2^n}$ and then argue by continuity ; but I would like a more direct proof . Thanks in Advance Is there a way of creating a filesystem object akin to this:  mknod files p cat file1 file2 ... fileN &gt; files   but such that it can be seeked in, as if it were a regular file? Is it appropriate to buy a "thank you" gift for a PhD supervisor? I am nearing the end of my PhD and I would quite like to buy a "thank you" gift for my supervisor (although, she's the socially awkward type who I'm sure will be greatly embarrassed to receive one).  I'm just wondering whether a gift is the done thing? And if so, what sort of gift seems appropriate? I suppose a bottle of wine would be a safe choice? Equaling 3 Displays Total (or 2, if the laptop display cannot be used).  I work at home on two large monitors, but at the office on a laptop with a single large monitor. Is it possible to attach two (or more?) external monitors to a laptop without having them clone each-others display?