Integer solutions $d = \frac{ab}{a + b + 2\sqrt{ab}}$
Eliminating non-integer solutions to $ab / (2\sqrt{ab} + a + b)$
I'm looking to create a simple short-lived reservation system, and I'd like to generate confirmation numbers that are   unique random-looking alphanumeric short-ish, at least much shorter than 32 character-long strings returned by sha1   I'm only looking to have ~500 reservations, so I don't imagine high likelyhood of collissions.  One idea I had is generate an sha1 hash based on a date-time stamp and username, then truncating it to its first 10 characters.  Would something like that be reliably unique enough for the purposes of processing ~500 reservations? a, b are integers. p is prime. I want to prove: $(a+b)^{p} \equiv a^p + b^p \pmod p$  I know about Fermat's little theorem, but I still can't get it I know this is valid: $(a+b)^{p} \equiv a+b \pmod p$ but from there I don't know what to do.  Also I thought about $(a+b)^{p} = \sum_{k=0}^{p}\binom{p}{k}a^{k}b^{p-k}=\binom{p}{0}b^{p}+\sum_{k=1}^{p-1} \binom{p}{k} a^{k}b^{p-k}+\binom{p}{p}a^{p}=b^{p}+\sum_{k=1}^{p-1}\binom{p}{k}a^{k}b^{p-k}+a^{p}$ Any ideas?  Thanks! Why does casting an ArrayList 's generic to a superclass not work? Why doesn't the mirror modifier work with object groups? Why doesn't the mirror modifier work with object groups? I have a group made of four objects to which I want to apply a mirror modifier. For me, it's not working. What are the steps to make apply a mirror modifier to a group of objects? I have 2 equation in terms of $a,b$ and $c$ .  $3a + 10b + 5c =0$ and $4a + 6b + 2c =0$  I need to find a:b:c and answer is $\dfrac a5 = \dfrac {b}{-7} = \dfrac {c}{11}$  I want to know how to get that?  My attempt:  Given equations can be written in form   $$\left ( \begin{matrix}           3 &amp; 10 &amp; 5 \\           4 &amp; 6 &amp;  2 \\           \end{matrix} \right )  \left ( \begin{matrix}           a\\         b\\         c\\         \end{matrix} \right ) =  \left ( \begin{matrix}            0 \\          0\\         \end{matrix} \right ) $$  But here I don't have any idea how to proceed. I can do for 3×3 matrix and I thought it can be done in same way.  Thanks. How to convert an MKV to AVI with minimal loss To convert an MKV to AVI, I do two things. The first thing I do is this:  ffmpeg -i filename.mkv -vcodec copy -acodec copy output.avi   or this:  ffmpeg -i filename.mkv -sameq -acodec copy output.avi   Either of these will convert the MKV to an AVI, but the problem is that the video does not play smoothly for some reason. That's fine though, because if I do one more thing it gets fixed:  ffmpeg -i output.avi -vcodec mpeg4 -b 4000k -acodec mp2 -ab 320k converted.avi   After I do this then the file plays without problem. I had success doing it this way for one file, but then I tried it on another file, and there is a slight, but noticeable loss in video quality. This is the output I get when doing the second step:  FFmpeg version 0.6.1, Copyright (c) 2000-2010 the FFmpeg developers   built on Dec 29 2010 18:02:10 with gcc 4.2.1 (Apple Inc. build 5664)   configuration:    libavutil     50.15. 1 / 50.15. 1   libavcodec    52.72. 2 / 52.72. 2   libavformat   52.64. 2 / 52.64. 2   libavdevice   52. 2. 0 / 52. 2. 0   libswscale     0.11. 0 /  0.11. 0  Seems stream 0 codec frame rate differs from container frame rate: 359.00 (359/1) -&gt; 29.92 (359/12) Input #0, avi, from 'output.avi':   Metadata:     ISFT            : Lavf52.64.2   Duration: 00:04:17.21, start: 0.000000, bitrate: 3074 kb/s     Stream #0.0: Video: mpeg4, yuv420p, 704x480 [PAR 229:189 DAR 5038:2835], 29.92 fps, 29.92 tbr, 29.92 tbn, 359 tbc     Stream #0.1: Audio: vorbis, 48000 Hz, stereo, s16 Output #0, avi, to 'converted.avi':   Metadata:     ISFT            : Lavf52.64.2     Stream #0.0: Video: mpeg4, yuv420p, 704x480 [PAR 229:189 DAR 5038:2835], q=2-31, 4000 kb/s, 29.92 tbn, 29.92 tbc     Stream #0.1: Audio: mp2, 48000 Hz, stereo, s16, 320 kb/s Stream mapping:   Stream #0.0 -&gt; #0.0   Stream #0.1 -&gt; #0.1   I just used arbitrarily large settings on the second step and it worked nicely before but not in this case. What settings should I use? If the distribution of test statistic is bimodal, does p-value mean anything? Are Jedi bound by a code of honor or morals? In The Phantom Menace alone, Qui-Gon (and in part Obi-Wan) are quick to use Jedi mind tricks and not only gamble but also cheat at it. They seem happy with this fact.  Do Jedi have morals and ethics, or a code of honor? I have Ubuntu server 12.04 installed, so I have no GUI. When I do the command ifconfig, I cannot find my internal IP address. It says inet addr: 127.0.0.1. Here is the output of ifconfig -a: eth0   link encap:Ethernet  HWaddr 00:06:4f:4a:66:f0     BROADCAST MULTICAST  MTU:1500  Metric:1     RX packets:0 errors:0 dropped:0 overruns:0 frame:0     TX packets:0 errors:0 dropped:0 overruns:0 carrier:0     collisions:0 txqueuelen:1000     RX bytes:0 (0.0 B)  TX bytes:0 (0.0 B)  eth1   link encap:Ethernet  HWaddr 00:16:ec:05:c8:9c      BROADCAST MULTICAST  MTU:1500  Metric:1     RX packets:0 errors:0 dropped:0 overruns:0 frame:0     TX packets:0 errors:0 dropped:0 overruns:0 carrier:0     collisions:0 txqueuelen:1000     RX bytes:0 (0.0 B)  TX bytes:0 (0.0 B)  lo     Link encap:Local Loopback     inet addr 127.0.0.1  Mask:255.0.0.0     inet6 addr:  ::1/128 Scope:Host     UP LOOPBACK RUNNING MTU:65536  Metric:1     RX packets:1800 errors:0 dropped:0 overruns:0 frame:0     Tx packets:1800 errors:0 dropped:0 overruns:0 carrier:0     collisions:0 txqueuelen:0     RX bytes:143896 (143.b KB)  TX bytes:143896 (143.8 KB)  here are the contents of etc/network/interfaces: # The loopback network interface auto lo iface lo inet loopback  # The primary network interface auto eth0 iface eth0 inet dhcp  If someone could edit this for me, the contents of etc/network/interfaces should be on separate lines. The output of host askubuntu.com was: ;; connection timed out; no servers could be reached.  I set up owncloud and webmin a few months ago and was using them for a month with no problems. I think the power went off one day 2 months ago and I never turned the server back on until yesterday. I haven't done anything that would have affected the internet setup So i'm not sure why it doesn't work anymore. As far as my network topology goes, I have a pci-e network card for the pc. The ethernet line goes from the network card to a switch, and then to a modem/router from there. Why would I want to use Interfaces? I understand that they force you to implement methods and such but what I cant understand is why you would want to use them. Can anybody give me a good example or explanation on why I would want to implement this. New top bar doesn't show hot threads across Stack Exchange sites Prevent Adobe Reader from switching to "Fit Page" zoom when bookmark is clicked Extending an enum via inheritance I know this rather goes against the idea of enums, but is it possible to extend enums in C#/Java?  I mean "extend" in both the sense of adding new values to an enum, but also in the OO sense of inheriting from an existing enum.    I assume it's not possible in Java, as it only got them fairly recently (Java 5?).  C# seems more forgiving of people that want to do crazy things, though, so I thought it might be possible some way.  Presumably it could be hacked up via reflection (not that you'd every actually use that method)?  I'm not necessarily interested in implementing any given method, it just provoked my curiosity when it occurred to me :-) My computer no longer boots Windows or the installed operating system. Instead, I may get this error:  PXE-E61: Media test failure, check cable -or- Check cable connection!  PXE-M0F: Exiting PXE ROM.   and/or one of the following messages or similar:  Operating System not found  No boot device available- No bootable devices--strike F1 to retry boot, F2 for setup utility  No bootable device -- insert boot disk and press any key  No Boot Device Found. Press any key to reboot the machine  Default Boot Device Missing or Boot Failed  Reboot and Select proper Boot device or Insert Boot Media in selected Boot device   What does this mean and what can I do?     This question comes up often and the answers are usually the same. This post is meant to provide a definitive, canonical answer for this problem. Feel free to edit the answer to add additional details. Taking a spin off of this question:  in a new direction for 2014.  Taking into account a basic Windows infrastructure:   domain controllers Exchange 2007/2010/2013 Sharepoint SQL File Servers / Print Servers AD Integrated DNS AD authenticated 3rd party devices (let's say 802.1X for networking and maybe some content-filtering, etc.) AD/LDAP authenticated "administrative" functions on IT apps/hardware/etc. perhaps some KMS stuff throw in a CA if you'd like home grown apps 3rd party in-house apps   Now, let's rip it all out and decide we are going to the cloud.  We've contracted to move Exchange/Sharepoint/File Services to Office 365.  SQL will now be hosted as well on  something like Azure.  We've gotten away from the need for AD-DNS and simply run everything via a simple Windows DNS server.  We still need 802.1X and would like SSO if possible to our various cloud apps.  Home grown and 3rd party in-house apps would likely stay, but have the ability to use internal user databases instead of AD authentication  The question is...do we really need Active Directory at all?  Or more to the point, AD on-premise or even hosted via Azure or similar (ADFS) or running ADDS on a hosted VM through Azure or similar.  Could/Should we look to something else like a 3rd party SSO option such as  or similar that can provide SSO functionality even if it is as simple as LastPass or similar for each user?  What kind of legitimate needs does AD fulfill if everything else in the cloud?  Could a MS-centric infrastructure get away with not having AD at all if they move everything that previously relied on AD to SaaS offerings that didn't rely on AD authentication? Does C# have anything comparable to Python's list comprehensions? Is GNOME Terminal a type of non-login shell? I have a hard time formulating proofs.  For this problem, I  can see that if $n$ is equal to $8,$ this statement is true.  $(\mathbb{Z}/8\mathbb{Z})^{\times}$ includes elements:  $1,3,5,7$, and all of these are roots of $1-x^2 \pmod 8.$  And obviously $8$ divides $24.$  But how do I prove this without depending on number calculations and only using theorems?  Help Please?  I need a step by step walk through of how to do this proof and what theorems would be appropriate to use. I have a motherboard with a PCIexpress x16 slot.  I want to purchase a card called AMD Radeon HD 7770, which uses PCIe version 3.0.  Is that backwards compatible? Please bring back the oldest sort order for answers, I think we can deal with four different types of sort order.  It's helpful if you don't want to have to tool-tip every post to see which got posted first, so that you can upvote the oldest answer when answers duplicate each other.  Edit:  Also, trying to read newest backwards doesn't work well for questions with multiple pages of answers.