Let $p \ge 5$ be prime. Prove $p^2 \equiv 1 \pmod {24}$. I'm having some trouble drawing anything useful from $p$ being prime. What can we say about p being prime that will help prove the statement?
Let $p&gt;3$ be a prime. Prove that $24 \mid p^2-1$.   I know this is very basic and old hat to many, but I love this question and I am interested in seeing whether there are any proofs beyond the two I already know.
What defines a AAA game? I tried to find a precise definition. I found clues on  and , but I cannot find more than an approximation like:     Equivalent of blockbuster movie in cinema, an AAA game is a game with highest development budgets and levels of promotion.   The definition seems imprecise. How can I be sure a game is AAA?   Let's imagine a small indie studio making games and growing up. They hire more and more people through the years and invest more and more money in their game. At which milestone would their game be considered as AAA? Pivot in SQL 2008 R2 How do the Hat Dash points work? Index notation Lorentz transfromation matrix Rings and some of their properties So I am studying for an exam and was wondering if anyone could explain some of the reasoning being the proofs of the following properties:   Let R be a ring with $a,b\in R$. Then    $a0=0a=0$  $a(-b)=(-a)b=-ab$ $(-a)(-b)=ab$   Proof 1:   $a0=a(0+0)=a0+a0$ hence $a0=0$   I don't really understand why $a0+a0$ proves that $a0=0$ This integral is very closely related to the sophmores dream that states  $$ \int_0^1 x^{-x}\mathrm{d}x = \sum_{n=1}^\infty n^{-n} = 1.27\ldots $$  For example here   Now I want to bound the integral, and showing that is less that 2.  For the interval $[0,1]$ a good bound is rewriting it to $\exp(x\log x)$ and using the expansion   $$ 1 - x \log(x) + \frac12 (-x \log(x))^2$$  but how does one handle $[1,\infty)$ ? In this answer here     gives bounds to the integral, but they are not tight enough.. So to taste my question again, how does one prove that   $$ \int_0^\infty \frac{\mathrm{d}x}{x^x} \leq 2 $$ How does one prove that addition on $\mathbb{R}$ is associative? If someone else is booking international ticket for me using his credit or debit card, does he need to be present physically at the time of check-in?   Suppose I am coming back from Singapore to my home country and my ticket is booked by my friend for me, who lives in my home country. Do airlines in Singapore ask for the person to be present physically to check in? How to check if file exists on FTP before FtpWebRequest C1/D Visa - Mexico and US Calculating $\lim_{n\to\infty}\sqrt{n}\sin(\sin...(\sin(x)..)$ How do I update Ubuntu manually? I wanted to update Ubuntu manually, because I used CDMA USB modem and connected using wvdial, but it won't able to update Ubuntu via Update Manager, because my Ubuntu won't showing "Connected" message via network-manager.  Can I update Ubuntu manually? Am I should download separate files, or use terminal instead? Integer reciprocals in arithmetic progression Let $m_1,m_2,\ldots,m_k$ be $k$ positive integers such that their reciprocals are in AP. Show that $k&lt;m_1+2$. Also find such a sequence.     Whatever way I tried, whichever formula I used, I could not eliminate the $m_k$ term, which I reckon would be the ideal scenario to find the solution. I am exhausted of any further ideas, please help. Thank you. java.rmi.NoSuchObjectException: no such object in table I am writing a very simple RMI server, and I am seeing intermittent java.rmi.NoSuchObjectExceptions in the unit tests.   I have a string of remote method calls on the same object, and while the first few go through, the later ones will sometimes fail. I am not doing anything to unregister the server object in between.  These error do not appear always, and if I put in breakpoints they tend to not appear. Are those Heisenbugs, whose race conditions dissolve when looking at them through the slowed down execution of the debugger? There is no multi-threading going on in my test or server code (though maybe inside of the RMI stack?).  I am running this on Mac OS X 10.5 (Java 1.5) through Eclipse's JUnit plugin, and the RMI server and client are both in the same JVM.  What can cause these exceptions? Economics.SE's favicon is missing in action What equipment should I have to kill the Eye of Cthulhu? I just manually summoned the Eye of Cthulhu, going into battle with copper armour, tungsten broadsword, gold bow, jester arrows and plenty of lesser healing potions. I came rather close to death but I managed to stay alive. However, the Eye took off when I had his health down to ~1700. I'm assuming this means you only have a few minutes to finish it off after a certain point. Is there any equipment I'm missing to help kill the Eye faster and to help my survivability? If it's relevant, my current life is 120. Class teardown in junit 3? I don't like Hell, can I go somewhere else? I just opened a nether portal for the first time, and I must say, it's not a very good location, a sort of narrow trench/cavern thingy without any obvious clues as to which direction is "out".  I'm not willing to simply move my overworld portal (due to dilation it is unlikely to have much of an effect, anyway), and was wondering if there was a way to force minecraft to generate a new personal Hell for me to explore (hopefully one that doesn't involve me stuck in some wall!). How do I make environment variable changes stick in Python? From what I've read, any changes to the environment variables in a Python instance are only available within that instance, and disappear once the instance is closed.  Is there any way to make them stick by committing them to the system?  The reason I need to do this is because at the studio where I work, tools like Maya rely heavily on environment variables to configure paths across multiple platforms.  My test code is  import os os.environ['FAKE'] = 'C:\\'   Opening another instance of Python and requesting os.environ['FAKE'] yields a KeyError.  NOTE: Portability will be an issue, but the small API I'm writing will be able to check OS version and trigger different commands if necessary.  That said, I've gone the route of using the Windows registry technique and will simply write alternative methods that will call shell scripts on other platforms as they become requirements. What points in the complex plane satisfy $|z|=arg(z)$.