I already proved it if $1729$ divides $a$, or if it doesn't divide $a$ but it's Greatest Common Divisor is not equal to $1$ (we get that the left side is equal to $0$ and the right side is equal to $0$ thus in these cases $a^{1729}\:=_{1729}\:\:a$ is a true statement). But I can't find a proof for $1729$ does not divide $a$ and it's Greatest Common Divisor is equal to $1$.
I want to know a proof of an alternative form of Fermat-Euler's theorem   $$a^{\phi (n) +1} \equiv a \pmod n$$   when $a$ and $n$ are not relatively prime.   I searched some number theory books and a cryptography book and internet, but there were only proofs of the original theorem  $a^{\phi (n)} \equiv 1 \pmod n$ or in case n=pq which is used for RSA. So I would be very thankful if someone show me a proof or book I should read.
Evaluating $\int_{-\infty}^{\infty}\frac{1}{(x^2+b^2)^2}dx$ Find $$\int_{-\infty}^{\infty}\frac{1}{(x^2+b^2)^2}dx$$   We see that the only poles are at $x=\pm bi$.  Integrating over the semicircular contour implies that it is equal to $2\pi i*Res_{(+bi)}$ because the integral over the circular arc is $0$.  So now I just need to calculate the residue at $+bi$. Can anyone give hints on how to do this?  I was thinking about multiplying by $(x-bi)^2$ and then evaluating at $bi$ but I'm not sure why this even makes sense. "I had my house [be] burned down" I have found out that using the verb be in passive constructions such as:      I had my house be burned down    is incorrect, therefore it should be      I had my house burned down.    But is it possible instead of be to use get? E.g.      The wire is passed through the pliers in a specific way to avoid having it get bent.   The rope had its surface wear.   The rope had its surface get worn.   The rope had its surface worn.   The first two sound very odd to me, but just for clarification could you please point out the errors and which ones are correct? Or is using get, be it in -ing or the to infinitive, in any situation along with have incorrect? What damage will powering down instead of shutting down do? How to manage a redirect request after a jQuery Ajax call What's the difference between "these" and "those"? First of all, I'm not a native English speaker, but in school I learned that these is used if referring to something near, and those is used when referring to something far away (temporally or locally). But now I'm sometimes watching English movies and notice sentences like "Have you seen those boots?" while the boots were only centimeters away. Is this just not correct, or have I learned the wrong rules?  Another question is, what if I'm referring to something abstract such as ideas or thoughts so that I can't say if they are far away or not. Should I use those or these?  Edit  So it can't be really wrong in any case if I use these respectively those? In the review page, it shows stats of recent reviewers. But it is showing wrong stats. Because today (on Monday) it shows week count 53 (on Meta Stack Overflow) which is impossible as there is a limit of 20 reviews per day.    PS: This bug   and . But the bug has occurred again. Which pseudo-$R^2$ measure is the one to report for logistic regression (Cox & Snell or Nagelkerke)? I have SPSS output for a logistic regression model. The output reports two measures for the model fit, Cox &amp; Snell and Nagelkerke.  So as a rule of thumb, which of these $R^²$ measures would you report as the model fit?  Or, which of these fit indices is the one that is usually reported in journals?    Some Background: The regression tries to predict the presence or absence of a bird (capercaillie) from some environmental variables (e.g., steepness, vegetation cover, ...). Unfortunately, the bird did not appear very often (35 hits to 468 misses) so the regression performs rather poorly. Cox &amp; Snell is .09, Nagelkerke, .23.    The subject is environmental sciences or ecology. How to check if value is in range of cells? I want to check if the value of a cell exists in a range of cells. How can I do this?  Just like you would be able to do "if value in list_of_values" in Python for example. I am currently writing my thesis and need to include a bunch of email conversions. I am using classicthesis. I wonder, how you would format stacked conversions. Moreover, how would you deal with the email header, signatures and line breaks: why do I need to add \\ all the time? This is the configuration I am currently experimenting with:  \documentclass[ oneside, openright, titlepage, dottedtoc, numbers=noenddot, headinclude, footinclude=true, cleardoublepage=empty, abstractoff, BCOR=5mm, paper=a4, fontsize=11pt, ngerman, american ] {scrreprt}  \PassOptionsToPackage{utf8}{inputenc} \usepackage{inputenc} \usepackage{blindtext}  \usepackage{framed}  % Vertical line that highlights quotations in an email. % Use \begin{leftbar} and \end{leftbar} environment. % Width of the line = 0.4pt % Space between the line and the text = 10pt % Requires \usepackage{framed} \renewenvironment{leftbar}{%   \def\FrameCommand{\vrule width 0.4pt \hspace{10pt}}%   \MakeFramed {\advance\hsize-\width \FrameRestore}}% {\endMakeFramed}  \usepackage{nicefrac} \usepackage[english]{babel} \usepackage[pdfspacing]{classicthesis}  \begin{document} \selectlanguage{ngerman} % american ngerman % Disable paragraph indentation. \setlength{\parindent}{0pt}     \begin{verbatim} From: john.doe@yahoo.com Subject: Your inquiry from 20042012 Date: Thu, 3 May 2012 11:34:54 +0200 To: Jane Doe &lt;jane.doe@hotmail.com&gt; \end{verbatim}  Jane Doe wrote: \begin{leftbar} \begin{leftbar} \begin{leftbar} \begin{leftbar} There are a couple of questions I like to ask. Please forward them if you cannot do so. \end{leftbar} I hope I can answer them after holidays. \end{leftbar} I am back from holidays now and will look through your questionaire the next days. \end{leftbar} Thank you, Mr. Doe - I am awaiting your answers. \end{leftbar}  Dear Mrs. Doe,\\ \ Lorem ipsum dolor sit amet, consectetur adipiscing elit. Ut molestie placerat pharetra. In quam leo, tincidunt ut consequat eget, malesuada sit amet urna. Morbi et interdum arcu. Nullam nisi nunc, ullamcorper vitae pulvinar eu, suscipit ut augue.\\ \ Donec risus mi, accumsan ac interdum id, molestie sed mauris. Suspendisse porttitor luctus justo in ullamcorper. Proin in neque magna. Mauris quis arcu eget enim mattis pharetra vitae a dolor. Nam aliquam consectetur mollis. Ut at sem quis dolor faucibus fringilla. Quisque cursus vehicula odio, vitae placerat ipsum dapibus id.\\ \ Best regards\\ John Doe\\ \ (See attached file: Questions\_Doe.doc)  \begin{verbatim} _______________________________________________________ Administration New York John Doe Mainstr. 3 11111 New York  T:        (011) 212-4301 F:            (011) 212-954301 M:         john.doe@yahoo.com \end{verbatim}  \begin{leftbar} My question I looking forward to reading the answers to my questions:\\ \ 1. Is the world analog or digital? \end{leftbar} digital  \begin{leftbar} 2. How long does the world exist?  \end{leftbar} analog since 1984, digital since 1999  \end{document}     This is what the above latex example looks like.    I also saw  but I do not want to use &gt; or colors. My settings are inspired by this  and this . How do I count the number of occurrences of a char in a String? Allow us to vote to delete from the Android app Currently, the only way to vote to delete (and undelete, since you can't see deleted posts) in the Android app is to open the post in a browser, and then vote to delete from there.  Can we get delete vote functionality within the app, as well as the ability to delete our own posts?  Worth to mention: users , so it's possible in the API.  Related: I would like to know if there is some way to share a variable or an object between two or more Servlets, I mean some "standard" way. I suppose that this is not a good practice but is a easier way to build a prototype.  I don't know if it depends on the technologies used, but I'll use Tomcat 5.5    I want to share a Vector of objects of a simple class (just public attributes, strings, ints, etc). My intention is to have a static data like in a DB, obviously it will be lost when the Tomcat is stopped. (it's just for Testing) When I look at the definition given from the Mac OS X Dictionary (I have set American English as interface language, and the dictionary used is then the New Oxford American Dictionary), I read:     the /ði/ /ð(ə)/ /ði/ [called the definite article]   adjective   1. denoting one or more people or things already mentioned or assumed to be common knowledge: what's the matter? | call the doctor | the phone rang. Compare with a.   • used to refer to a person, place, or thing that is unique: the Queen | the Mona Lisa | the Nile.   • informal denoting a disease or affliction: I've got the flu.   • (with a unit of time) the present; the current: dish of the day | man of the moment.   • informal used instead of a possessive to refer to someone with whom the speaker or person addressed is associated: I'm meeting the boss | how's the family?   • used with a surname to refer to a family or married couple: the Johnsons were not wealthy.   • used before the surname of the chief of a Scottish or Irish clan: the O'Donoghue.   I have never thought of an article as adjective. Is normally an article defined as adjective? Radiation– white vs black house, hot or cool? How to enable default file manager(nautilus) to handle desktop icons Is there a way to check the size of the video memory? Specifically, is there one that works accurately for both integrated GPU's as well as PCI/AGP graphics cards?  Many integrated GPU's have dynamically allocated memory, so the solution would hopefully return either the maximum available video memory or the currently allocated amount. For stand-alone NVidia or ATI cards it would obviously return the total amount of physical GPU RAM.  lspci -v does output memory figures, but I do not believe it is the video memory. I suspect the figure reported is some system memory allocation or block or channel size, but I don't know for sure. You can see in these  that lspci was wrong in 5 of the 6 tests:  ** ASUS EN210 PCIe - 1024 Mb ***   01:00.0 VGA compatible controller: nVidia Corporation GT218 [GeForce 210] (rev a2)         Subsystem: ASUSTeK Computer Inc. Device 8354         Memory at e3000000 (32-bit, non-prefetchable) [size=16M]         Memory at d0000000 (64-bit, prefetchable) [size=256M]         Memory at e0000000 (64-bit, prefetchable) [size=32M]  *** Galaxy 8400GS PCIe - 512 Mb ***   01:00.0 VGA compatible controller: nVidia Corporation G98 [GeForce 8400 GS] (rev a1)     Subsystem: nVidia Corporation Device 05cc     Region 0: Memory at e4000000 (32-bit, non-prefetchable) [size=16M]     Region 1: Memory at d0000000 (64-bit, prefetchable) [size=256M]     Region 3: Memory at e2000000 (64-bit, non-prefetchable) [size=32M]  *** VirtualBox VM - 10 Mb (headless server) ***   00:02.0 VGA compatible controller: InnoTek Systemberatung GmbH VirtualBox Graphics Adapter         Memory at e0000000 (32-bit, prefetchable) [size=16M]  *** VirtualBox VM - 128 Mb ***   00:02.0 VGA compatible controller: InnoTek Systemberatung GmbH VirtualBox Graphics Adapter (prog-if 00 [VGA Controller])         Memory at e0000000 (32-bit, prefetchable) [size=128M]  *** S3 Savage 4 AGP - unknown Mb (old lspci log), but I don't think they made these cards with 128Mb memory! ***   00:01.0 VGA compatible controller: S3 Inc. Savage 4 (rev 06) (prog-if 00 [VGA controller])     Subsystem: IBM Unknown device 01c5     Region 0: Memory at feb80000 (32-bit, non-prefetchable) [size=512K]     Region 1: Memory at f0000000 (32-bit, prefetchable) [size=128M]  *** NVIDIA Quadro FX 1800 integrated - 1024 Mb ***   01:00.0 VGA compatible controller: nVidia Corporation GT215 [Quadro FX 1800M] (rev a2) (prog-if 00 [VGA controller])     Subsystem: Dell Device 040c     Memory at e2000000 (32-bit, non-prefetchable) [size=16M]     Memory at d0000000 (64-bit, prefetchable) [size=256M]     Memory at e0000000 (64-bit, prefetchable) [size=32M] Dynamic pivot: sum of sales per month My table is like this:  Sitecode    Month  Amount --------    -----  ------ XX          Jan     1000 XX          Jan     3000 XX          Apr     3000 XX          Apr     1000   What I want is, to show the result something like this:  Sitecode    MonthJAN   MonthAPR --------    --------   -------- XX          4000       4000 Has any Dark Force user/ally ever said "May the Force be with you"? How can I check if my computer's powerful enough to run a certain game? How to create a fancy table of contents for Beamer How can I create a fancy-looking table of contents for a Beamer presentation?    \documentclass{beamer} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc}  \begin{document}  \frame{ \tableofcontents }  \section{Foo bar} \frame{} \section{Foo bar} \frame{} \section{Foo bar} \frame{} \section{Foo bar} \frame{} \section{Foo bar} \frame{} \section{Foo bar} \frame{} \section{Foo bar} \frame{}  \end{document}