I am curious about the nature of the primes in  $$ \frac{m(m+1)(m+2)...(m+k-1)}{k!} $$  when $k &lt; m$ ( means they dont overlap).  We can show that it is always an integer using n choose k formulation or pascals rule or by prime factor counting.  But this last proof got me thinking. How can it be shown that there will be other primes as factors than those in $k!$ ?  Edit: I want to show that there is at least one prime greater than k as factor in that in every case.  I am looking for a non-handwavy yet insightful proof.  Edit2: I want to show that there is at least one ( specific form not needed ) prime greater than k that divides the product of any k consecutive natural numbers. just existence proof.   In other words, I want to show that there is always a prime greater than k between k and n/2 or between n-k and n, whenever n is strictly greater than 2k.
Sylvester's theorem states that the product of $k$ consecutive integers $&gt;k$ has a prime factor $p&gt;k$. Erdős gave a short proof for it, but I can't find Erdős's proof online. Can anyone give a sketch of the proof? Thanks.
Run cmd.exe in Administrator mode without explorer.exe I've recently had trouble with explorer.exe not working.  I found a possible solution , using .  The problem is that I cannot "right-click Command Prompt, and then click Run as administrator", since explorer.exe does not work for even a second without crashing.  I tried to use the  to open cmd as administrator, but I get an error 1327 because I have a blank password.    My question is, is there any other way to run cmd.exe with admin privileges?  If not, is there any way to run sfc.exe without admin privileges?  If not, does anyone have solution to my original problem of explorer.exe stopping and restarting ad nauseam? When $\operatorname{Hom}_{R}(M,N)$ is finitely generated as $\mathbb Z$-module or $R$-module? Assume that $M$ and $N$ are two finitely generated $R$-modules. Then $\operatorname{Hom}_{R}(M,N)$ is a finitely generated $\mathbb Z$-module and/or $R$-module (in this case, assume that $R$ is commutative)?   Note that $R$ is not Noetherian ring. Is there any counterexample (for any cases)? Apologies if this has been asked somewhere before, but I didn't see what I was looking for after several pages of Google results.  I was reading Jech's The Axiom of Choice and was introduced to the equivalence of the ultrafilter lemma and the compactness theorem for FOL. I gather that these are also equivalent to the completeness theorem. I am primarily interested in trying to understand the "ultrafilter lemma $\implies$ completeness/compactness" direction. Jech's own proof takes an odd route through "binary messes" and I don't really follow the last couple of steps. (It's on pp.17,18 of the 2008 Dover reprint, if one is interested).  I'm wondering if anyone knows a good reference for, or can provide, a readable proof that the ultrafilter lemma implies completeness or compactness, with a preference for completeness. Alternatively, if you have Jech's book on hand, helping me bridge the concluding paragraph of his $\textrm{(ii)}\implies\textrm{(iii)}$ step with the preceding steps would clear up much of my puzzlement. (Though in this case I can't offer any help in saying what, specifically, is missing in the way of a bridge). Package Listings Warning: Text dropped after begin of listing when calling function I'm trying to this:  \newcommand{\code}[1] {\begin{lstlisting} {#1} \end{lstlisting} }   But I keep getting this error:  Package Listings Warning: Text dropped after begin of listing   I'm calling \usepackage and all so, do any of you knows how to solve this? On webmail every user gets error "ERROR: Message not sent. Server replied: 354" while sending mails.   But mails are sent properly and reaches properly.  Can you tell me where is problem, and why it is occurred?  UPDATE: I found in some resources that this is not an error but standard information message. On the other side, it's annoying. Can you tell me if it is a true, and if it is how can I avoid it? Gauss law question with regard to this example I am really confused in Gauss law.  Why do E3 and E2 pointing up? and also E1 pointing down?  The lecture note said infer from symmetry and you will get the following but I dont really understand. Could anyone explain to me? I am new to gauss law. Launch iOS app from sites Can we add functionality to the mobile site to view the page in the SE iOS app? It could be through  or a link at the bottom where the download link is.        I really don't care which, but right now there's just no way to do it. Paying off student loan or using that money for a downpayment on a house Drawing automata for languages I'm trying to draw two automata for these two languages:   For the first one, I know that the minimum is n = 1, m = 1, but I'm having troubles drawing a NFA for it.  The second one the minimum is n = 2, m = 1, but I still don't know how to start the NFA.  I have this for q2: biblatex custom reference I've been searching for a way to create "Custom References" with biblatex, but haven't found a solution to my problem. Among "normal sources" like Journal Articles, I need other references where I can put a custom sentence in it. Like "[3] This was performed by someone else".  Is there an option to do that?   \documentclass{scrreprt} \usepackage[refsection=chapter,style=chem-angew]{biblatex} \usepackage[english]{babel} \usepackage[T1]{fontenc} \usepackage[ansinew]{inputenc} \begin{filecontents}{biblio.bib}  @article{entry1,      author = "George D. Greenwade",     title = "The {C}omprehensive {T}ex {A}rchive {N}etwork ({CTAN})",     year = "1993",     journal = "TUGBoat",     volume = "14",     number = "3",     pages = "342--351",            url=" www.ctan.org" }  \end{filecontents}  \addbibresource{biblio.bib}  \begin{document} \chapter{Test Chapter One} \cite{entry1} So far so good, now I need another reference that says 'This was performed by someone else'. \printbibliography[heading=subbibliography]  \end{document} How to prove this result involving the quotient maps and connectedness? Given topological spaces $X$ and $Y$, where $Y$ is connected, let $p \colon X \to Y$ be a quotient map. If, for each point $y \in Y$, the set $p^{-1}(\{y\})$ is connected, then how to prove that $X$ is connected also?   By the map $p$ being a quotient map is meant the following:   The map $p$ is surjective, and, for any subset $V$ of $Y$, the set $V$ is open in $Y$ if and only if $p^{-1}(V)$ is open in $X$. Why is usable RAM less than total RAM? My girlfriend bought a laptop last week. It's a core 2 duo with 4 GB  We installed vista 64bit, and one of the first things we did was right click on "My computer" to see gthe properties. Immediately we noticed something strange about her RAM, the line said: Installed memory (RAM): 4,00 GB (3,68 GB usable)  I told her not to worry, thinking it must be something about the laptop hardware (considering her vista installation came from the same DVD as mine, and I never noticed anything like that on my 4 GB desktop). One hour ago, it got worse. We looked at Properties again, and it now says: Installed memory (RAM): 4,00 GB (2,98 GB usable)  What does that mean? Are those 1,02 GB missing or being used by the system?  EDIT: There is a possibility that the sytem information is wrong. I just noticed that it reports an intel T6500 processor, when it's actually a T6400.  How can I find out how much RAM is really available to the system?  EDIT2: Checking the resource monitors, it says 1003 MB are reserved for the hardware. Is that good or bad? Thanks I just try to visualize some points using GeoServer to publish them as WMS and bind this WMS to my Openlayers website using the GeoServer internal Heatmap process to visualize it. I don't want to use the Openlayers Heatmap function.   But there seem to be some problems, look here:    I'm using this Openlayers implementation to bind my WMS URL:    Changing the params to:  params: {'LAYERS': 'test:points','TILED': true,'STYLES': 'heatmap'},   And using this as SLD to define the Heatmap:    Any ideas why this happens? strtok segmentation fault Prove every element of $G$ has finite order. Let $G$ be a group such that the intersection of all its subgroups which are different from $\{e\}$, is a subgroup different from $\{e\}$. Prove that every element of  $G$  has finite order.  Assume that $a$ has infinite order in $G$. Then let $y\neq e$ belong to the intersection of all subgroups of $G$ different from $\{e\}$. Then $y=a^n$ and $y$ will also belong to subgroup generted by $a^2$ hence  $y=a^{2m}$ for some integers $m$. This contradicts that $a$ has infinte order.  Is this correct? How do you search for backdoors from the previous IT person? We all know it happens. A bitter old IT guy leaves a  into the system and network in order to have fun with the new guys and show the company how bad things are without him.  I've never personally experienced this. The most I've experienced is somebody who broke and stole stuff right before leaving. I'm sure this happens, though.  So, when taking over a network that can't quite be trusted, what steps should be taken to ensure everything is safe and secure? I have answered 'No' to Siri initial prompt, but I can still see it in my process list. I've tried to kill it, but it loads again immediately. How can I unload and remove Siri from my Mac? Getting hybrid graphics to work nvidia-prime GT650M I recently made the switch to Ubuntu. Everything on my laptop works fine except my hybrid graphics card. It is Intel graphics combined with Nvidia GeForce GT650M. I tried Bumblebee, but primusrun doesn't work (segmentation fault, core dumped) and optirun works with glxspheres64, but steam games look worse with optirun than with Intel graphics (e.g. Surgeon Simulator 2013 runs fine with Intel and hangs a lot with optirun)   I tried Nvidia Prime, but it gives me a black screen on startup after install, so I can't do anything and have to power off my PC with the power button. I also tried nvidia-prime 0.5 on Ubuntu 14.04, and then I get 'low-graphics mode' and I can't get to Unity. How to import an FBX file into Blender I am trying to import an FBX file from Unity into Blender. Does anybody know how to do that? When using QuickLook to quickly look through multiple images (pressing space and then using the arrow keys to navigate), some images will not get displayed by QuickLook when hitting them up a second time (going back for example).   A GIF does a much better job than me showcasing the issue. So here's me previewing a few random images:    I have tried deleting   ~/Library/Preferences/com.apple.QuickLookDaemon.plist ~/Library/Caches/com.apple.QuickLookDaemon/Cache.db ~/Library/Caches/com.apple.QuickLookDaemon32/Cache.db ~/Library/Containers/com.apple.quicklook.ui.helper  and restarting, unfortunately to no avail.  Repairing permissions and restarting the Finder in 32-bit mode, as suggested  by others, don't appear to have any effect on this issue either.  This happens with all kinds of images, no matter their source or format.