Prove that, for every natural number $n &gt; 2$, there is a prime number between $n$ and $n!$. [Hint: There is a prime number that divides $n! - 1$.]
How to prove that there exists a prime number between $n$ and $ n!$, for all $ n&gt; 2$?    ( gives a much better bound, but this question is about obtaining a self-contained proof.)
Missing Schengen entrance stamp Last night I took a night train from Romania to Hungary. I shared a compartment with another American backpacker (and others.) The border crossing on the Romanian side was normal -- all of our passports were stamped and the border officer recorded our data in his notepad. On the Hungarian side, some sort of commotion broke out further down the train car while we were being processed. The officer who was scanning and stamping our passports left and never came back. As a result, the other American backpacker never had her passport scanned or stamped.  What sort of difficulties might she run into since her passport has not been recorded as entering the schengen zone, nor does she have an extra stamp? What might get options be now, from within Schengen, to get her entrance recorded?  (When I say "scanned", the officers had hand-held devices that read the passport biometrics and I assume, recorded some sort of data somewhere.) Can't operator == be applied to generic types in C#? A deck of 52 cards is shuffled, and we deal a bridge of 13 cards. Let $X$ be the number of aces and $Y$ be the number of spades. Show that $X$, $Y$ are uncorrelated.  Here is what I did:  $Cov(X,Y) = E[XY]-E[X]E[Y]$   uncorrelated means $Cov(X,Y) = 0$, hence  $E[XY]=E[X]E[Y]$   $E[X] =  \sum_{k=0}^{k=4} k \frac{\dbinom{4}{k} \dbinom{48}{13-k}}{\dbinom{52}{13}} $   $E[Y] =  \sum_{k=0}^{k=13} k \frac{\dbinom{13}{k} \dbinom{39}{13-k}}{\dbinom{52}{13}} $   Are the summations above correct? and how do I calculate $E[XY]$? Is there some guide I can use for cooking soft boiled ostrich eggs? No Indian (and Bengali) keyboard layout in Ubuntu 17.04? I've upgraded from Ubuntu 16.10 to 17.04 today. I'm from Bangladesh, and Bengali is my native language. There was an option to set keyboard layout to Bengali in all previous Ubuntu versions I've used. But there's no such option in 17.04, strangely. I can use Bengali by installing the package ibus-m17n,  so I thought maybe there was some problem during my upgrade. But then I checked my friend's PC which runs ubuntu 16.10 and found that ibus-m17n was not installed in 16.10 but there's option for Bengali keyboard layout. So is Bengali keyboard support really removed from 17.04, or was there a problem during my upgrade? How to add vias and more traces? How can I change Win 10 protected registry entries? If I take the $k \in \mathbb{N}$ power of 10 and mod it by a large prime, I notice that the remainders repeat at some point.  For instance $10^k mod~7$ seems to repeat every $8$th value of $k$.  Given $k$ (the power of ten), and a large prime number $p$, how can I compute the size of the cycle?  I tried writing a python script that computes the first instance of a repeat. However, I don't know if this works since a single instance of a repeat may not necessarily mean a full cycle. Correlations between continuous and categorical (nominal) variables How to ssh into a server running on a home network, behind a router? Okay, so I have a PC in my home network, running Ubuntu on a VM with openssh server installed.    How can I remotely access it from another computer outside the home network, using the internet? What IP address do I use to connect to it? What kind of port-forwarding do I need to do? Browsing through a copy of Counterexamples in Analysis (Gelbaum  &amp; Olmsted), I'm quite enjoying the typeface used for 'script' letters (see links to examples below); visually it's somewhere between what I'd now call calligraphic and fraktur, and semantically it's used where now we'd use mathcal, mathfrak or mathbb.  I was therefore procrastinatiously looking for a LaTeX-usable font to include these characters, but couldn't seem to find any. I'm certain they're not unique to this text or publisher, so figured my lack of success was probably more to do with searching with the wrong keywords. If anyone has any ideas I'd be grateful. So, let's take Skeptics.SE for an example. You can become a trusted user when you acquire 4000 rep., not 20000  I was like, wow, so is it like this because of the lack of users on Skeptics.Se, as in, not so much as Stack Overflow or something?  Why do some sites need less reputation to be a trusted user? Decomposition of semisimple Lie algebra via its roots Exercise 14.33 in Fulton and Harris's Representation Theory claims that  If the roots of a semisimple Lie algebra lie in a collection of mutually orthogonal subsets, one sees that the Lie algebra decomposes into a direct sum accordingly.  Although intuitively I can see why this is true, could anybody direct me to a proof of this assertion, or outline how I should go about doing it? AmpScript to select Right or Left characters from a string Is there an AmpScript function that can be used in an equivalent manner to sql left, right and mid functions?   Thanks, Mark Difference between modelAttribute and commandName attributes in form tag in spring? How to make terminal autocomplete when there are several files/directory? If I have several directories, like:  afoo abar   sometimes my terminal will refuse autocomplete when I press tab (e.g. "cd a" then tab), and print the list of directories instead. Sometimes it even throws a noisy, annoying sound. Any idea how to make it autocomplete in cases like this? E.g it can show abar first, and then afoo if I press tab again. I saw this is the case in windows, or some applciation in Ubuntu How do I check if an array includes a value in JavaScript? What is the most concise and efficient way to find out if a JavaScript array contains a value? This is the only way I know to do it: function contains(a, obj) {     for (var i = 0; i &lt; a.length; i++) {         if (a[i] === obj) {             return true;         }     }     return false; }  Is there a better and more concise way to accomplish this? Why directory with large amounts of entries does not shrink in size after entries are removed? Abstraction VS Information Hiding VS Encapsulation Can you tell me what is the difference between abstraction and information hiding in software development?  I am confused. Abstraction hides detail implementation and information hiding abstracts whole details of something.  Update: I found a good answer for these three concepts.  for several citations taken from . I'm trying to prove whether or not the Laurent polynomials $\mathbb{Z}[x, x^{-1}]$ with coefficients in $\mathbb{Z}$ form a principal ideal domain.    I know that $\mathbb{F}[x, x^{-1}]$ is a PID when $\mathbb{F}$ is a field, but clearly $\mathbb{Z}$ is not a field so I cannot appeal to this result. And my intuitions are not serving me very well at the moment. Can anyone provide a hint or direction to take?