What can we say about the number of generators in group $\mathbb{Z}^*_p$ ($p$ prime)?
Cyclic Group Generators of Order $n$
$L^p$ and $L^q$ space inclusion Let $(X, \mathcal B, m)$ be a measure space.  For $1 \leq p &lt; q \leq \infty$, under what condition is it true that $L^q(X, \mathcal B, m) \subset L^p(X, \mathcal B, m)$ and what is a counterexample in the case the condition is not satisfied? Differences between /bin, /sbin, /usr/bin, /usr/sbin, /usr/local/bin, /usr/local/sbin Which Content Management System (CMS)/Wiki should I use? This is a general, community wiki  question to address non-specific "I need a CMS or Wiki that does x, y, and z..." questions.  If your question was closed as a duplicate of this question and you feel that the information provided here does not provide a sufficient answer, please open a discussion on .    I have a list of features that I want for my website's Content Management System (CMS) - where can I find a [free] script that includes all of them? How many files can a windows folder contain?  This question is asked  but then the question text pollutes the question with issues of "rules of thumb"  So to be clear, this question is: How many files can a windows folder contain? Is there a single word for "one who speaks/boasts a lot about everything"? I'm looking for a single word to most aptly describe a person possessing the following "qualities":   Appears to be superior in every technology/skill under the sun, which he deems worthy of knowing On being questioned/assigned the task, is found to have only superficial/no knowledge of the said skill. Inherent tendency to "one-up" anyone and everyone - example: "I didn't work hard at all in my undergrad days, but yet I have better/equal grades than you have! "   Could someone help?  Edit: I've got some very good responses, but despite popular opinion for "braggart", I'm going with "pretentious" as it suits what I was looking for more aptly IMHO. See code:   var file1 = "50.xsl"; var file2 = "30.doc"; getFileExtension(file1); //returns xsl getFileExtension(file2); //returns doc  function getFileExtension(filename) {     /*TODO*/ } Proving $\, a^b = (a\bmod n)^b\, \pmod{\!n}\ $ [Congruence Power Rule] I am working on a problem I am pretty close to solving but I can't figure out the last part. I used some algebraic manipluation to break the problem down. The problem is:     Show that the following congruence is true      a^b = ((a mod n)^b) (mod n)      for any positive integers a, b, and n.      Hint: Number a can be written as a = q*n + r, where q = floor brackets a/n, and r = a mod n.   My attempt it:     (q*n +r)^b = ((a mod n)^b) * (mod n)      (q*n)^b + (a mod n)^b = ((a mod n) ^b) * (mod n)      (a)^b -(r)^b = ((a mod n) ^b) * (mod n)      (a)^b = modn  ----> divided by ((a mod n)^b)   Am I doing this correctly and if so how can I prove this is true? Is there a trick I am missing? I think that by simplifying it further I would just keep going in a loop.  Thanks! I've got a repo that I contribute to within an Organization that I administer. I'm trying to add it to careers, but it won't allow it. Is this on purpose? I don't really want to fork it to my personal account, since I won't be maintaining it from there. How do you set a default value for a MySQL Datetime column?  In SQL Server it's getdate(). What is the equivalant for MySQL? I'm using MySQL 5.x if that is a factor. What's the easiest way to set up a LAMP stack? I set up a new VPS instance of Ubuntu and am wondering what the easiest way is to get up and running with a basic LAMP stack (i.e. which packages are required, which configuration options need to be tweaked, if any, etc.). How do I recover my Google account (or Gmail) password or username? I have forgotten the password or username for my Google account, or I'm unable to access my account for other reasons.  How can I recover my account?  (This Q&amp;A is meant as a general description for all Google and Gmail username/password questions. See: ) How do we "restore" notepad after crash? How do I force TeX to completely use the 1st column before writing to the second column?  Right now I dont have enough content to fill one whole page. So the left and right columns are equally filled, leaving 25% symmetrical empty page at bottom. How do I fill the left column first, and then the right?  \documentclass{article}  \usepackage{multicol} \usepackage[scale=0.9]{geometry} \usepackage{amssymb} \renewcommand{\labelitemi}{\tiny$\blacksquare$} \begin{document} \begin{multicols}{2}      %lots of text goes here  \end{multicols} \end{document} Following an earlier question ,  I would love to implement the  as suggested by Russell McMahon.   That way I could switch between 4AA batteries (1.2V*4 or 1.5V*4) and USB power. The battery voltage would nicely be regulated to 5V too. And the idea was to use the shutdown pin of the chip IF USB power is detected. The device also has to work when no batteries are present.  I am not too experienced with this, but I thought to connect the 5V from the USB to the SHUTDOWN pin from the LDO chip. Pitywise, the pin function is inverted, if SHUTDOWN is low, Vout from the LDO chip is low.  I am now considering using an XOR gate to make the thing work:         &ndash; Schematic created using   I would love to limit the amount of components, but the number of diodes is already high, and the whole thing feels a bit crafted   My questions:    Is there a better suitable chip (with an inverted shutdown pin) is there a better circuit (without voltage drop) to get this (I would prefer to get rid of D1, but then, the XOR has no desired operation) What are the in universe telltale indicators of an away beam down or beam up? What do witnesses experience? As a long time fan, I'm familiar with the tell tale sound effects that indicate when the beaming process is taking place. What do residents of the Star Trek universe witness from the ground when people beam back to or down from the ship? Does GCM (or GHASH) only provide 64-bit security against forgeries? In a  a doubt was voiced about my answer, which claims GCM to requires $2^{128}$ for a successful forgery. The doubt was that the square root needs to be taken meaning the security would be $2^{64}$.  So of course I immediately checked the relevant security theorem ():  $$\mathbf{Adv}^{\text{auth}}_{\operatorname{GCM}[\operatorname{Perm}(n),\tau]}(\mathcal A)\leq \frac{0.5(\sigma+q+q'+1)^2}{2^n}+\frac{q'(\ell_A+1)}{2^\tau}$$  with $\sigma$ being the total plaintext size in blocks, $q$ being the total number of encryption queries, $q'$ being the total number of decryption queries, $n$ being the size in bits of the underlying permutation, $\ell_A$ being maximum authenticated input length in blocks and $\tau$ being the tag size in bits.  Now clearly from this we can see that performing $2^{n/2}=2^{64}$ queries yields a strong enough advantage for a break.  Now my question is: When talking about "n-bits of security" or "needs $2^n$ operations to break", do we usually talk about "online" security, ie queries to an oracle can be done and cost $1$ or "offline" security where no oracles are available or do we need to make this decision depending on the situation?  TL;DR: Does GCM offer 64 or 128 bit security? Short story: people rediscover how to do math and want to put people in bombs Many years ago I remember reading a short story about an advanced civilization at war that rediscovers how to do math mentally. The reason I remember it is the guy tells it to government officials who are delighted because it means they can put people inside bombs or missiles instead of computers, which were thought to be more valuable.  I believe this is one of those stories I read in a science fiction anthology my father brought home on one of his business trips in the sixties or seventies. Non-Exchange address in Exchange GAL We just made the switch to Google Apps for part of our domain. We disabled the accounts that are now on Google Apps and mail routes correctly. Our problem is that now we have no way of seeing those users in the Global Address List.  So we have these users in our Active Directory, they have no mailbox on Exchange, and we want to be able to see/search for their addresses in the GAL. Is there any way to do this?  This is Exchange 2007, if that matters. SQL injection that gets around mysql_real_escape_string() For which Natural $n\ge2: \phi(n)=n/2$