I was wondering if there are any $n$ that for $ \frac {2^{n+1}-1}{n+1}$ this number is natural, so:  $$k=\frac {2^{n+1}-1}{n+1}      \qquad k,n \in \mathbb{N}                 $$  let $n+1=t$  $$ k\times t=2^{t}-1              $$  Now we have that $k\;$and $\;t$ are odd, so $n\;$ is even. I tried to rewrite it into form $k,t=2a+1$, but it didn't work. So i'm curious is there even a solution to this problem because i made it by myself
Here is a problem which i have not been able to do for quite sometime.   For $n \geq 2$, show that $n \nmid 2^{n}-1$.   I have thought of proving this in two ways: One by using induction which didn't actually work. Next by Fermat's little theorem we have $2^{p-1} \equiv 1 \ (\text{mod} \ p)$, which actually says that for $p \mid 2^{p}-2$. But i couldn't proceed more than this. Any ideas by which i can actually solve the problem?
Avoiding Error 1325 EsriPyFldr not valid short file name when uninstalling ArcGIS for Desktop? I am trying to uninstall ArcGIS 10.3.1 for Desktop from Windows 7 SP1 using Uninstall a program and I am receiving an error message (pictured below):     Error 1325. EsriPyFldr is not a valid short file name.   The error message dialog appears just after:     Computing space requirements   seems to complete.  If anyone else has encountered this, then have you been able to find a way to get past it and uninstall ArcGIS 10.3.1 for Desktop?  My machine has what I would call a vanilla install of ArcGIS 10.3.1 for Desktop and ArcGIS Pro 1.1.1 so there are Python versions 2.7.8 and 3.4.1 as a result. ArcGIS Earth Beta is also on the machine.     I have made a posting on the same issue to   After running this utility () unsuccessfully (it hung for two hours before I killed it) the error message when trying to uninstall shifted to:     This action is only valid for products that are currently installed     ArcMap 10.3.1 continues to start fine and doing  wmic product where "Vendor like '%Environmental Systems Research Institute%'" get Name, Version, InstallDate, InstallLocation in a DOS window (at the suggestion of @MattWilkie) reports only the presence of Python 3.4.1 for ArcGIS Pro, ArcGIS Pro and ArcGIS Earth on my machine. The question Is there any way to automatically scale and put an entire document on an A0 page? The thesis for which this is to be done is available as LaTeX code, as well as PDF output. Of course I know it is possible to do this manually, but I would prefer a LaTeX solution.  A small background Our university introduced a new page limit for master theses. This is not exactly appreciated by the teaching staff as well as students. To make a little joke, we decided to try to arrange an entire thesis (of approximately 200 pages) on a A0 poster, put it up in the hallway, and put a magnification glass next to it with a note saying: &quot;New page limit introduced: 1 page maximum for all upcoming theses!&quot;  The final result As some of you asked for this: below you can see a picture of the final result. (I &quot;professionally removed&quot; some of the sensitiv information.) We presented the poster during my final presentation and everyone liked it a lot! Now, it is on display next to my office and is still attracting some attention once in a while. Script to change current directory (cd, pwd) I want to run a script to simply change the current working directory:  #!/bin/bash cd web/www/project   But, after I run it, the current pwd remains unchanged! How can I do that? Apollonius circle, its radius and center Mapping JSF .xhtml files to no extension How does the radius of a pipe affect the rate of flow of fluid? Poiseuille's law states that the rate of flow of water is proportionate to $r^4$ where $r$ is the radius of the pipe. I don't see why.  Intuitively I would expect rate of flow of fluid to vary with $r^2$ as the volume of a cylinder varies with $r^2$ for a constant length. The volume that flows past a point is equal to rate of flow fluid, and thus it should vary with $r^2$.  I cannot understand the mathematical proof completely. Is there an intuitive explanation for this? The Raspberry Pi B+ models have a protection circuit between the USB connector and the 5V net on the board.  They recommend putting a similar protection circuity on a Pi HAT before 'backpowering' the pi through its GPIO header along with a polyfuse.  I understand why this is the recommendation, but I'd like to understand more about how this circuit works.  I did some searching before posting this question and found information on using a MOSFET as a low voltage drop diode, but they all had the gate wired directly to ground without the pair of PNP's and the resistors.  What are they doing for this circuit?  Also, is this primarily using the body diode?  In which case, what's the relevant info on the datasheet that qualifies DMG2305UX for this application?  In the other circuits I found, it appeared low Rdson and Vgsth compatible with the circuit seemed like the relevant characteristics. For some complicated diagrams, I need to use something like Adobe Illustrator or Inkscape, and would like to include nicely typeset labels. (I'm aware of post-processing solutions, such as pinlabel.) One way to do this is to use a closely-matching font, and then convert fonts to outlines. Better, however, would be to use Computer Modern directly in Illustrator.  Where can I find a Computer Modern font compatible with Adobe Illustrator? Convolution is uniformly continuous and bounded Suppose $f\in L^\infty(\mathbb{R})$ and $K\in L^1(\mathbb{R})$ with $\int_\mathbb{R}K(x)dx=1$. Show that the convolution $f\ast K$ is a uniformly continuous and bounded function.  The definition of the convolution is $(f\ast K)(x)=\int_\mathbb{R}f(x-y)K(y)dy$.   There is the inequality $\|f\ast K\|_\infty\leq\|f\|_\infty\|K\|_1$, which yields that $f\ast K$ is bounded. But what about uniform continuity? Read echo'ed output from another PHP file When do I use a dot, arrow, or double colon to refer to members of a class in C++? Perl - Package/Module Issues Proving a distance between molecules defines a metric space. A DNA molecule can be represented as a string of symbols $A$, $C$, $G$ and $T$, such as  $$GGATAATTCTAG. . .GACCGTACCC$$ For the purposes of this question, we will assume that all DNA molecules contain the same (large!) number $N$ of symbols. Thus, a DNA molecule is an $N$-tuple $x = (x_1, . . . , x_N)$ where $x_i  ∈ \{A,C,G,T\}$ for each $i$.  Define the distance between two DNA molecules $x$, $y$ as the number of elements $i ∈ {1, . . . ,N}$ such that $x_i \neq y_i$. Prove that this defines a metric on the set of DNA molecules. Etymology of Tor and Ext Functors The names of the derived functors $\operatorname{Tor}$ and $\operatorname{Ext}$ seem quite cryptic to me. Does anyone know what these abbreviations stand for? I would be glad if someone could tell me where these names come from. Can't save my render as an image I seem to be having issues when I try and save the output file as a .png, or .jpeg for that matter. I've set up my user preferences, File, Render Output to my desktop as well as render output in the editor window. What am I missing here? I'm obviously very new to this program and your help would be appreciated. Thanks for your time Zorn's Lemma states that if every chain $C$ in a partially ordered set $X$ has an upper bound then there is at least one maximal element in $X$.  Let $n\in \mathbb{N}$, $C_n \subset \mathbb{N}$ be the set containing the natural numbers $\{1,\dots,n\}$. Then if $k\le n$, $C_k\subset C_n$, so $C_n$ is a chain. Also, $C_n$ has an upper bound, such as $n+1$. But what is the maximal element of $\mathbb{N}$?  There's something I don't get in Zorn's Lemma statement. Can I add an exterior outlet by tapping an adjacent interior outlet? Can I tap in a outlet not being used that is behind a bed and add a outlet behind it outside? Using it for a electric bbq grill. The grill specs are: 110 volt, 60hz, 275w, 5 amp service. Diamond moderator rollbacks can resurrect blacklisted tags how to install the DDE to the ubuntu 20.04? I have installed the Ubuntu 20.04, and want now I want to install Deepin Desktop Environment (dde)  My steps are:   add the ppa and install using apt  sudo add-apt-repository ppa:leaeasy/dde sudo apt-get update sudo apt install dde    But during the command sudo add-apt-repository ppa:leaeasy/dde I get the following error:  $ sudo add-apt-repository ppa:leaeasy/dde  You can install deepin desktop environment by run  sudo add-apt-repository ppa:leaeasy/dde sudo apt-get update apt-get install dde ------ Ubuntu Bionic: https://drive.google.com/file/d/1xkQPb4tWocknuYeY-t41W--kz-9SSL0V/view?usp=sharing ------ Important Issue: Q: Can dde on Ubuntu 16.04? A: Sorry, the dde depends on qt version above than 5.6.      If you want to install dde, please upgrade to 17.04 or above.  More info: https://launchpad.net/~leaeasy/+archive/ubuntu/dde Press [ENTER] to continue or Ctrl-c to cancel adding it.  Hit:1 http://in.archive.ubuntu.com/ubuntu focal InRelease Hit:2 http://in.archive.ubuntu.com/ubuntu focal-updates InRelease                                      Hit:3 http://in.archive.ubuntu.com/ubuntu focal-backports InRelease                                    Ign:4 http://ppa.launchpad.net/leaeasy/dde/ubuntu focal InRelease                                      Hit:5 http://security.ubuntu.com/ubuntu focal-security InRelease            Err:6 http://ppa.launchpad.net/leaeasy/dde/ubuntu focal Release             404  Not Found [IP: 91.189.95.83 80] Reading package lists... Done E: The repository 'http://ppa.launchpad.net/leaeasy/dde/ubuntu focal Release' does not have a Release file. N: Updating from such a repository can't be done securely, and is therefore disabled by default. N: See apt-secure(8) manpage for repository creation and user configuration details.   I get the similar error with sudo apt-get update :  $ sudo apt-get update Hit:1 http://in.archive.ubuntu.com/ubuntu focal InRelease Hit:2 http://in.archive.ubuntu.com/ubuntu focal-updates InRelease                                       Hit:3 http://in.archive.ubuntu.com/ubuntu focal-backports InRelease                                     Ign:4 http://ppa.launchpad.net/leaeasy/dde/ubuntu focal InRelease            Hit:5 http://security.ubuntu.com/ubuntu focal-security InRelease Err:6 http://ppa.launchpad.net/leaeasy/dde/ubuntu focal Release   404  Not Found [IP: 91.189.95.83 80] Reading package lists... Done E: The repository 'http://ppa.launchpad.net/leaeasy/dde/ubuntu focal Release' does not have a Release file. N: Updating from such a repository can't be done securely, and is therefore disabled by default. N: See apt-secure(8) manpage for repository creation and user configuration details.   How can I install dde ? Show intersection of two algebras are not a $\sigma$-algebra I have the following question:    $\textbf{Question}:$ Let $\mathcal{F}_1$ and $\mathcal{F}_2$ be two algebras. Is $\mathcal{F}_1 \cap \mathcal{F}_2$ a $\sigma$-algebra?  I believe the answer is no. I know the definition of an algebra and the definition of $\sigma$-algebra.    I'm assuming I need to show that the intersection of the finite unions of each algebra is finite, but then I get stuck trying to work out how to show it is not a $\sigma$-algebra.  All help is appreciated.  Many thanks, John