I tried to play around with $2^{2^{n + 1}} - 1$, such as splitting it to $2^{2^{n} \cdot 2} - 1$, but it didn't helped much. I am kind of stuck and don't know what to do next
This is just a curiosity. I have come across multiple proofs of the fact that there are infinitely many primes, some of them were quite trivial, but some others were really, really fancy. I'll show you what proofs I have and I'd like to know more because I think it's cool to see that something can be proved in so many different ways.  Proof 1 : Euclid's. If there are finitely many primes then $p_1 p_2 ... p_n + 1$ is coprime to all of these guys. This is the basic idea in most proofs : generate a number coprime to all previous primes.  Proof 2 : Consider the sequence $a_n = 2^{2^n} + 1$. We have that  $$ 2^{2^n}-1 = (2^{2^1} - 1) \prod_{m=1}^{n-1} (2^{2^m}+1), $$  so that for $m &lt; n$, $(2^{2^m} + 1, 2^{2^n} + 1) \, | \, (2^{2^n}-1, 2^{2^n} +1) = 1$. Since we have an infinite sequence of numbers coprime in pairs, at least one prime number must divide each one of them and they are all distinct primes, thus giving an infinity of them.  Proof 3 : (Note : I particularly like this one.) Define a topology on $\mathbb Z$ in the following way : a set $\mathscr N$ of integers is said to be open if for every $n \in \mathscr N$ there is an arithmetic progression $\mathscr A$ such that $n \in \mathscr A \subseteq \mathscr N$. This can easily be proven to define a topology on $\mathbb Z$. Note that under this topology arithmetic progressions are open and closed. Supposing there are finitely many primes, notice that this means that the set $$ \mathscr U \,\,\,\, \overset{def}{=} \,\,\, \bigcup_{p} \,\, p \mathbb Z $$ should be open and closed, but by the fundamental theorem of arithmetic, its complement in $\mathbb Z$ is the set $\{ -1, 1 \}$, which is not open, thus giving a contradiction.   Proof 4 : Let $a,b$ be coprime integers and $c &gt; 0$. There exists $x$ such that $(a+bx, c) = 1$. To see this, choose $x$ such that $a+bx \not\equiv 0 \, \mathrm{mod}$ $p_i$ for all primes $p_i$ dividing $c$. If $a \equiv 0 \, \mathrm{mod}$ $p_i$, since $a$ and $b$ are coprime, $b$ has an inverse mod $p_i$, call it $\overline{b}$. Choosing $x \equiv \overline{b} \, \mathrm{mod}$ $p_i$, you are done. If $a \not\equiv 0 \, \mathrm{mod}$ $p_i$, then choosing $x \equiv 0 \, \mathrm{mod}$ $p_i$ works fine. Find $x$ using the Chinese Remainder Theorem.   Now assuming there are finitely many primes, let $c$ be the product of all of them. Our construction generates an integer coprime to $c$, giving a contradiction to the fundamental theorem of arithmetic.  Proof 5 : Dirichlet's theorem on arithmetic progressions (just so that you not bring it up as an example...)  Do you have any other nice proofs?
Generic htaccess redirect www to non-www I would like to redirect www.example.com to example.com. The following htaccess code makes this happen:  RewriteCond %{HTTP_HOST} ^www\.example\.com [NC] RewriteRule ^(.*)$ http://example.com/$1 [L,R=301]   But, is there a way to do this in a generic fashion without hardcoding the domain name? What is a good way to transform Cyclic Ordinal attributes? I can describe "IMHO" as an acronym for "in my humble opinion."  How can I describe the reverse processes of translating the acronym into the actual phrase?  That is to say, if "IMHO" is the acronym, what do we call "in my humble opinion"? Are any fictional worlds available in some GIS format? Is there any fictional world (e.g. ,  etc.) which could be used in a GIS context?  For example, an MBTiles archive and a shapefile or any other format usable with OpenLayers / Polymaps / Leaflet? What would be a good way to go about attacking a Quagmire 3 cipher?  I understand that it is polyalphabetic with a key and an indicator. I have started by taking the cipher text, taking every n-th letter and putting that in a string, computing the index of coincidence for the n strings I get and then averaging that. I would assume that whatever n value corresponds with the closest average index of coincident to English (0.066) would be the number of alphabets, i.e. the length of the indicator.  Where do I go from there? Or is my approach incorrect? Calculating the force required to lift a weight with a screw Can I connect a PC to a PoE switch? Groups of order $pq$ without using Sylow theorems If $|G| = pq$,  $p,q$ primes, $p \gt q, q \nmid p-1 $, then how do I prove $G$ is cyclic without using Sylow's theorems? Why do we need to dummy code categorical variables How can I assign a Func conditionally between lambdas using the conditional ternary operator? What is the easiest way to resolve apt-get BADSIG GPG errors? Some people use a term for a popular special case in place of a generic term.  (Often this popular special case is a particular product in that category.)  I think that this is a common phenomenon.  Is there any word/phrase for it?  For example, “Coke” officially refers to a  (I think), but some people seem to call any carbonated soft drinks “Coke.”  In Japanese, famikon is the Japanese name for , a video game console which was extremely popular in 1980s and early 1990s, but some Japanese people call any video game console famikon even if it is not really a famikon.  (As an aside, calling a video game console famikon when it is not really a famikon is even a stereotype of a “mother unfamiliar with technology” in Japan.)  I am looking for a term which refers to this kind of usage of words. How to check if a point is inside a rectangle? Expected maximum of sub-Gaussian I'm trying to answer the following question from the book :  Let $X_1,X_2,\dots$ be a sequence of sub-gaussian random variables, which are not necessarily independent. Show that   $E\bigg[ \max_i \frac{|X_i|}{\sqrt{1 + \log i}} \bigg] \le CK$,  where $K = \max_i \|X_i\|_{\psi_2}$. Deduce that for ever $N \ge 2$ we have   $E\bigg[ \max_{i \le N} |X_i| \bigg] \le CK \sqrt{\log N}$.  I've tried to figure out what is the distribution of the maximum of Gaussians, but I'm reaching only inequalities that that don't help me answer the question.   I've also seen a similar question .  Does anyone have a clue or something to start with in order to answer this question?  Thanks! Order of items in a context menu Can we make the "expanded user card" privilege universal? Pursuant to discussion on the , several have asked how far they'll need to search to determine if a user has expressed their preferred pronouns. This requested feature could create an easy solution for that concern.  On sites where I've reached the  privilege level, users can hover over my picture to see a snippet of my profile:    This is where I've included my preferred pronouns, and where I see many others including them as well.  By making this feature universal, users could quickly and easily check for preferred pronouns without needing to navigate to another page. GridBox/packed array behaviour Here is my situation: I know almost nothing about Perl but it is the only language available on a porting machine. I only have permissions to write in my local work area and not the Perl install location. I need to use the  Perl module from CPAN   How do I use this Parallel::ForkManager without doing a central install? Is there an environment variable that I can set so it is located?  Thanks  JD Positive integer multiples of an irrational mod 1 are dense I'm not sure how to solve this one. Thank you!  $2.$ For any $\alpha\in \mathbb R$ we define $$\lfloor \alpha \rfloor = \max_{n\in\mathbb Z}\{\,n\mid n\leq \alpha\,\}$$ and $$\alpha\bmod 1 = \alpha - \lfloor \alpha \rfloor$$ Let $\alpha$ be irrational. (a) Given $n\in\mathbb N$ show that $\{\,k\alpha\bmod1\mid k\in\mathbb N\,\}\cap\left[0,\frac{1}{n}\right]\neq\emptyset$ (b) Prove that $\{\,n\alpha\bmod 1\mid n\in\mathbb N\,\}$ is dense in $[0,1]$. Suppose I have a 6 GB page file on an SSD and another 6 GB page file on a traditional mechanical 5400rpm SATA drive.  Does Windows take into account the difference in the performance characteristics between the two drives when deciding which page file to use?  If it matters, I'm asking about Windows 7 and higher.  I reviewed this  but it doesn't appear to factor in the case of dissimilar storage speeds.