We say that a positive integer is full of squares,if in its factorization all the primes appear raised to an exponent greater than one. For example, $72 = 2^3⋅3^2$ is full of squares and $18 = 2⋅3^2$ is not. Prove that there exist infinite positive integers $n$ such that $n$ and $n + 1$ are filled with squares.   The only thing I think that I figured out is that the solution uses induction.
One way to prove that there is an infinity of consecutive powerful integers is to just note that there are infinite solutions to the equation $x^2-8y^2=1$, because clearly $x^2$ and $8y^2$ are clearly powerful. But this uses a bit of theory.  I found this problem under an induction worksheet, and all of the surrounding problems are easy. Is there really an elementary solution to the problem?  All of my attempts have been fruitless.  Note: a number is powerful if whenever a prime $p$ divides it we have that $p^2$ also divides it.
I know that for continuous variable $P[X=x]=0$.   But i can't visualize that if $P[X=x]=0$, there is infinite number of possible $x$'s. And also why do their probabilities get infinitely small ? Suppose f(m,n) is a double sequence in $\mathbb R$. Under what condition do we have $\lim\limits_{n\to\infty}\sum\limits_{m=1}^\infty f(m,n)=\sum\limits_{m=1}^\infty \lim\limits_{n\to\infty} f(m,n)$? Thanks! Is there any good reason that an empty set of round brackets (parentheses) isn't valid for calling the default constructor in C++?  MyObject  object;  // ok - default ctor MyObject  object(blah); // ok  MyObject  object();  // error   I seem to type "()" automatically everytime. Is there a good reason this isn't allowed? I am hearing my screen I am developing a game with a Minecraft-like terrain made out of blocks. Since basic rendering and chunk loading is done now, I want to implement block selecting.  Therefore I need to find out what block the first person camera is facing. I already heard of unprojecting the whole scene but I decided against that because it sounds hacky and isn't accurate. Maybe I could somehow cast a ray in view direction but I do not know how to check the collision with a block in my voxel data. Of course this calculations must be done on the CPU since I need the results to perform game logic operations.  So how could I find out which block is in front of the camera? If it is preferable, how could I cast a ray and check collisions? When I press the power button a dialog opens, but I want to shutdown the system without receiving "Do you really really want to shutdown?". How can I choose a direct shutdown action on a pressed power button?   I had the same problem with the 11.04 version, but I don't remember the solution and yesterday I upgraded to Ubuntu 11.10. I setup my Nginx conf as per  , and now http2 is available...  But in Chrome (Version 54.0.2840.98 (64-bit))  Dev tool, it's always on HTTP 1/1 :  NAME             METHOD  STATUS  PROTOCOL shell.js?v=xx..    GET    200     http/1.1   my server is running Ubuntu 16.04 LTS which supports both ALPN &amp; NPN , and the openssl version shipped with it is 1.0.2g  I checked http2 support with and the result is :    Yeah! example.com supports HTTP/2.0. ALPN supported...   Also checking with curl is OK   $ curl -I --http2 https://www.example.com   HTTP/2 200    server: nginx/1.10.0 (Ubuntu)   date: Tue, 13 Dec 2016 15:59:13 GMT   content-type: text/html; charset=utf-8   content-length: 5603   x-powered-by: Express   cache-control: public, max-age=0   etag: W/"15e3-EUyjnNnyevoQO+tRlVVZxg"   vary: Accept-Encoding   strict-transport-security: max-age=63072000; includeSubdomains   x-frame-options: DENY   x-content-type-options: nosniff   I also checked with is-http2 cli from my console  is-http2 www.amazon.com × HTTP/2 not supported by www.amazon.com Supported protocols: http/1.1  is-http2 www.example.com ✓ HTTP/2  supported by www.example.com Supported protocols: h2 http/1.1   tested with openssl from my localhost  $ echo | openssl s_client -alpn h2 -connect www.example.com:443 | grep ALPN  depth=2 O = Digital Signature Trust Co., CN = DST Root CA X3  verify return:1  depth=1 C = US, O = Let's Encrypt, CN = Let's Encrypt Authority X3  verify return:1  depth=0 CN = example.com  verify return:1  ALPN protocol: h2  DONE   why Chrome is left behind it ? How can I check it also with Safari (v 10.0.1) Proof that $n\Bbb Z \leq \Bbb Z$ and are the only subgroups of $\Bbb Z$ My challenge is      Prove that if $n = 0,1,2,\ldots$ and $n\Bbb Z = \lbrace nk: k \in \Bbb Z \rbrace$, show that $n\Bbb Z$ is a subgroup of $\Bbb Z$ and are the only subgroups.   I handled the first point by saying that  i) Identity is 0, and $0\in n\Bbb Z$ and $0 \in \Bbb Z$  ii) For any $a,b \in n\Bbb Z$ $a+b = na_2 + nb_2 = n(a_2+b_2) \in n \Bbb Z$  iii) For any $na \in n\Bbb Z$, its inverse $n(-a) \in n\Bbb Z$.  For the second point, I said that all of the subgroups should be infinite, because for any finite group the largest element $a+a \notin H$, where $H$ is the potential subgroup, which is a contradiction. So $H$ must be of infinite order. For any $a \in H, -a$ should also be in $H$. The difference between any $a,b \in H$ should also be in $H$, i.e. $(a-b) \in H$. This leaves only the whole group or any group of the form $n\Bbb Z$.  Is my proof complete? How to move a window if titlebar is overlapped by Ubuntu menu Somehow the title-bar of chrome was overlapped by Ubuntu menu-bar like attached image.  In this case how to move this chrome window? Glucagon stimulates glycogenolysis and gluconeogenesis, thus increasing the plasma glucose concentration — so that tissues get enough glucose in the fasting state. However glucagon also inhibits glycolysis. In this case, what is the point of having high blood glucose if it cannot be catabolized? I would like to have multiple bibliographies (reference lists) as described in   ().   However, I'd like to use with a single .tex file, which would contain the bibliography database embedded, as in  ...   Now, I want to use letter documentclass, for which the bibliography needs to be defined ()   So what I end up is with this example (code below, compiled with just pdflatex, no bibtex(which I'd like to avoid in this case)):     ... obviously, something works (multibib recognized primary and secondary cites, assigning them number "1" as start) -- however I don't get two, separate primary and secondary list of references; would that be possible in this context? (doesn't have to be with multibib, but I'd like to avoid a separate .bib file and a bibtex call)  EDIT: I'd also like separate cite keys (i.e. rendered as [a1] and [b1]; not yet sure if multibib can do that; edit: no it cant,  says "You can also edit the .bbl files created by BibTEX and add 'a', 'b',. . . to the labels, but your changes will be overwritten by subsequent BibTEX runs." and recommends \cite[a]{key}) - how possible would that be?  &nbsp;  The code is this:   \documentclass{letter}  % Environment thebibliography undefined when using letter: http://www.latex-community.org/forum/viewtopic.php?f=4&amp;t=3359 \makeatletter \setlength \labelsep {.5em} \newcommand\newblock{\hskip   .11em\@plus.33em\@minus.07em} \let\@openbib@code\@empty \newcommand\refname{References} \newcommand\section{\@startsection   {section}{1}{\z@}%   {-3.5ex \@plus -1ex \@minus -.2ex}%   {2.3ex \@plus.2ex}%   {\normalfont\Large\bfseries}} \newenvironment{thebibliography}[1] {\section*{\refname}%   \@mkboth{\MakeUppercase\refname}{\MakeUppercase\refname}%   \list{\@biblabel{\@arabic\c@enumiv}}%   {\settowidth\labelwidth{\@biblabel{#1}}%     \leftmargin\labelwidth \advance\leftmargin\labelsep \@openbib@code     \usecounter{enumiv}%     \let\p@enumiv\@empty \renewcommand\theenumiv{\@arabic\c@enumiv}}%   \sloppy \clubpenalty4000 \@clubpenalty \clubpenalty   \widowpenalty4000%   \sfcode`\.\@m} {\def\@noitemerr {\@latex@warning{Empty       `thebibliography' environment}}%       \endlist} \makeatother  \usepackage{multibib} \newcites{pri}{Primary sources} \newcites{sec}{Secondary sources}  \begin{document}   \begin{letter}{Addressee} Blah, blah - content...  Primary cite: \citepri{lamport94} ...  Secondary cite: \citesec{lamport94} ...   \begin{thebibliography}{9}  \bibitem{lamport94}   Leslie Lamport,   \emph{\LaTeX: A Document Preparation System}.   Addison Wesley, Massachusetts,   2nd Edition,   1994.  \bibitem{lamport95}   Leslie Lamport,   \emph{\LaTeX: A Document Preparation System}.   Addison Wesley, Massachusetts,   2nd Edition,   1995.  \end{thebibliography}  \end{letter}   \end{document} How to prove that $E[Y]=E[E[E[Y|X_1, X_2]]]$ ?  PS. I don't see how $E[E(Y|X_{1},X_{2})|X_{1}]=Y[Y|X_{1}]$ and $E[Y]=E[E(Y|X_{1})]$ can be used here. But it feels close. Please help, I'm stuck PPS. Thanks @Deep North. Let's consider the case when X1 is independent of X2 How to detect using c# if a pendrive is plugged into a USB port? Is there a way to find out when in a LAN anyone plugs in a pendrive to the USB port?  Programatically (in C# preferably) or through some tool. Basically I'd imagine a client application sits on each terminal and monitors the USB ports and sends the information to the server.   a.) Can I get the details of the file(s) being copied? b.) Is there a way to do this without a client application?   EDIT  I dont want to disable the USB port entirely. its to be on a need to have basis. Basically just want the users on the LAN to share data responsibly and know that whatever data is tranfered is monitored and logged and can be questioned later. I need to view the saved WiFi passwords on my non-rooted Android phone I have an LG Optimus 7 (not-rooted), and all my WiFi passwords are there. I want to view the saved WiFi passwords to transfer thm to my new Samsung Galaxy S3, but I don't know how.  I tried to search on the Internet and Google play for an application that would allow me to view this information, but I can't find one. Is there any way for me to accomplish this? What does "can only heal so much" mean in this sentence? I found the following sentence in an American novel (this one:  you can read the whole paragraph). A woman broke her wrist and just had her cast removed. She thinks she can start playing piano again, but her wrist still hurts too much. I'm not sure I understand the meaning of the final sentence:  But a body can only heal so much, so fast.  Can someone help me, please? How does one obtain a Witcher 3 Troll Mutagen from a non-quest specific troll, at least one of which you never exchange dialogue with? I have played the Witcher 3, exchanged dialogue with 5 trolls already, and decided not to kill them. All 5 trolls were involved in specific quests. I have killed two trolls of which I did not exchange dialogue. I have yet to receive a Troll Mutagen.  How does one obtain a Witcher 3 Troll Mutagen from a non-quest specific troll, at least one of which you never exchange dialogue with? What is the use case for a Fragment with no UI? Layers with same coordinates not lining up I have a project where I have a few layers (State of Texas, cities, roads, etc.). They are all resaved and projected to NAD83 / UTM zone 14N EPSG: 26914. The project CRS is also NAD83 / UTM zone 14N EPSG: 26914 and on the fly projection is enabled. I added a text delimited file and cannot get them to line up. I checked the attribute table of the delimited file and the coordinates are correct. I cross-checked this with a point close to where a few of the points from the delimited file are and they are very close so in theory they should be lining up. I tried to save the delimited file and save it with the same projection and add the shp file back in and that did not work. Anyone have any ideas? I can't think of anything else to try. I have a DB Server running CentOS 6.4 and MySQL 5.5 from RPMs (not source).  I have dozens of databases but one of them is being hit quite hard with large selects which locks out other updates which in turn locks out other selects (at least I think this is what's happening!)  To alleviate this I would like to replicate one of the databases onto the same server so that the writes can write to the "master" and reads can be done on the "slave". I don't have the luxury of additional hardware.  I have found a lot of guides on the internet describing how to replicate to a different server, and one () describing how to replicate to the same server but it doesn't seem to be geared towards retrofitting this functionality to an existing database with 10s of GB of data in it already. It also talks about compiling MySQL from source.  Does anyone know how I can retrospectively convert my MySQL 5.5 config to replicate one of my databases on the same server?  My existing DB runs on port 3306 so I feel it should be as easy as firing up another instance of MySQL on port 3307 and making some changes to my.cnf but not sure what changes.  Thanks in anticipation. Is there a way to run Python on Android?