Prove that $(x^n + y^n)$ is divisible by $(x + y)$ if $n$ is odd. Prove that $(x^n + y^n)$ is divisible by $(x + y)$ if $n$ is odd.   I don’t know how to proceed. Help please.
Proof of $a^n+b^n$ divisible by $a+b$ when $n$ is odd I read somewhere that     $(a^n - b^n)$         It is always divisible by $a-b$.   When $n$ is even it is also divisible by $a+b$.   When $n$ is odd it is not divisible by $a+b$.      and     $(a^n + b^n)$         It is never divisible by $a-b$.   When $n$ is odd it is divisible by $a+b$.   When $n$ is even it is not divisible by $a+b$.      I wonder what's the proof for this.  First postulate is clear. $(a-b)$ would or would not be a factor. Any light on others?
How to run my own program without specifying its path Let's suppose I have compiled something and I run it like so:  $ /path/to/my/executable/mycmd Hello World   What do I need to do to run it like   $ mycmd Hello World    from everywhere in my computer? Does $\int_{ - 1}^1 {\frac{1}{x}dx} = 0 \text { or } \infty $? Add an SQLite database to an iPhone app Every Christmas, Harry receives a mediocre gift from the Dursleys. In Harry Potter and the Chamber of Secrets he gets a toothpick.     At that moment, Hedwig swooped into the room, carrying a very small package in her beak.     “Hello,” said Harry happily as she landed on his bed. “Are you speaking to me again?”     She nibbled his ear in an affectionate sort of way, which was a far better present than the one that    she had brought him, which turned out to be from the Dursleys. They had sent Harry a toothpick    and a note telling him to find out whether he’d be able to stay at Hogwarts for the summer    vacation, too.    They hate owls, but obviously use them to contact Harry at school? Set root domain record to be a CNAME I need to create an NS record for a domain that is a CNAME, for the purpose of having two domains pointed at one IP, and not having to maintain the current IP address in two different places.  The DNS provider for this domain is DynDNS, but they block this operation:     CNAME cannot be created with label   that is equal to zone name   I can do this with another domain whose DNS is served by 1and1:  root@srv-ubuntu:~# dig myseconddomain.co.uk  ; &lt;&lt;&gt;&gt; DiG 9.4.2-P1 &lt;&lt;&gt;&gt; myseconddomain.co.uk ;; global options:  printcmd ;; Got answer: ;; -&gt;&gt;HEADER&lt;&lt;- opcode: QUERY, status: NOERROR, id: 61795 ;; flags: qr rd ra; QUERY: 1, ANSWER: 2, AUTHORITY: 0, ADDITIONAL: 0  ;; QUESTION SECTION: ;myseconddomain.co.uk.             IN      A  ;; ANSWER SECTION: myseconddomain.co.uk.      71605   IN      CNAME   myfirstdomain.co.uk. myfirstdomain.co.uk.     59      IN      A       www.xxx.yyy.zzz  ;; Query time: 298 msec ;; SERVER: 10.0.0.10#53(10.0.0.10) ;; WHEN: Tue Aug 18 14:17:26 2009 ;; MSG SIZE  rcvd: 78   Is this a breach of the RFCs or does DynDNS have a legitimate reason for blocking this action?  Followup Thanks to the two answers already posted I now know that 1and1 IS breaching RFCs to do this. However it does work and they seem to support it. For a company that hosts so many domains it seems very odd that they get away with doing this on such a massive scale without objection.  More followup  The output of "dig myseconddomain.co.uk ns" as requested.  root@srv-ubuntu:~# dig myseconddomain.co.uk ns  ; &lt;&lt;&gt;&gt; DiG 9.4.2-P1 &lt;&lt;&gt;&gt; myseconddomain.co.uk ns ;; global options:  printcmd ;; Got answer: ;; -&gt;&gt;HEADER&lt;&lt;- opcode: QUERY, status: NOERROR, id: 18085 ;; flags: qr rd ra; QUERY: 1, ANSWER: 2, AUTHORITY: 0, ADDITIONAL: 2  ;; QUESTION SECTION: ; myseconddomain.co.uk.             IN      NS  ;; ANSWER SECTION: myseconddomain.co.uk.      4798    IN      NS      ns67.1and1.co.uk. myseconddomain.co.uk.      4798    IN      NS      ns68.1and1.co.uk.  ;; ADDITIONAL SECTION: ns67.1and1.co.uk.       78798   IN      A       195.20.224.201 ns68.1and1.co.uk.       86400   IN      A       212.227.123.89  ;; Query time: 59 msec ;; SERVER: 10.0.0.10#53(10.0.0.10) ;; WHEN: Wed Aug 19 12:54:58 2009 ;; MSG SIZE  rcvd: 111 mkdir error: no such file or directory on a linux machine running CENTOS v 7.8.2003 I am in a directory Neutrinos I now do  mkdir /pionloop/ this works. I then go to into this new directory pionloop and do mkdir /E_0.3 and this works as well. I thus have as a result a directory Neutrinos/pionloop/E_0.3.  Now, starting in Neutrinos I want to do this in one command and therefore do mkdir /pionloop/E_0.3 and get: `mkdir: cannot create directory ‘/pionloop/E_3.0’: No such file or directory  What is going wrong here?? What is the function of "do" in this sentence? Rarely, though, do countries introduce a single reform alone.   What function does do have in this sentence? Hyperspace travel is a norm in Star Wars universe. Even small ships have that capability and without this capability galactic-scale economy isn't possible.  My educated estimates say that at any point of time at least billions of ships are in hyperspace across the galaxy. Now, talk about economic hubs like Coruscant during Clone Wars Era. The number of ships arriving here should be enormous. Millions in an hour? If one person small ship can have warping capability, then the number can exceed even billions in a peak hour. You can go shopping on a neighbor planet five times a day or you can live on a cheap planet even if you have a job on Coruscant (there are public transport cabs featured in Rebels TV show in case you aren't rich enough to have own ship).  Now, the famous quote:     "Traveling through hyperspace ain't like dusting crops, boy! Without precise calculations we could fly right through a star or bounce too close to a supernova, and that'd end your trip real quick, wouldn't it?"   That's fine. Calculations prevent you from hitting a star, supernova or planet which are already on map, but how does a ship avoid hitting another ship? Provided that number of ships arriving near a planet is that much high, the probability of two ships jumping to the same co-ordinate can't be zero. In fact, the probability should be very very high.  Is there a central body which controls the traffic by coordinating the hyperspace calculations of different ships? This seems unlikely because everyone seems to go anywhere they want while maintaining the destination privacy (there are countless examples). Or, is there really a traffic control body and those who travel privately actually gamble? Is there any other protocol if not traffic control body?  Has the canon ever addressed it? What about Legends? OpenLayers Plugin Error: latitude or longitude exceeded limits I'm aware the 'traceback' OpenLayers plugin error has been dealt with before, so apologies for raising the issue once again. Unfortunately for me, the solutions posted were way over my head, e.g. using workarounds and code posted on github, etc. I find it all mind-boggling.  If anyone could post a step by step answer for an absolute beginner like me I would be eternally grateful! Thank you. I hope that one day I can contribute something.  The error message reads:   An error has occured while executing Python code:  Traceback (most recent call last):   File "C:/Users/MCMA1580/.qgis//python/plugins\openlayers_plugin\openlayers_plugin.py", line 48, in addLayer     self.__plugin.addLayer(self)   File "C:/Users/MCMA1580/.qgis//python/plugins\openlayers_plugin\openlayers_plugin.py", line 190, in addLayer     self.__setMapSrsGoogle()   File "C:/Users/MCMA1580/.qgis//python/plugins\openlayers_plugin\openlayers_plugin.py", line 249, in __setMapSrsGoogle     extMap = coodTrans.transform(extMap, QgsCoordinateTransform.ForwardTransform) QgsCsException: forward transform of (0.884883, 2.549197) PROJ.4: +proj=longlat +datum=WGS84 +no_defs +to +proj=merc +a=6378137 +b=6378137 +lat_ts=0.0 +lon_0=0.0 +x_0=0.0 +y_0=0 +k=1.0 +units=m +nadgrids=@null +wktext  +no_defs Error: latitude or longitude exceeded limits I'm unable to access the following website from my Internet Explorer 8 under Windows XP with Internet Explorer unable to display web page error. By the other hand the FireFox browser at the same Windows XP can access this website.  How to solve the issue with IE8 ? Showing that $e^x > 1+x$ Lens upgrade paths (sub $1000) for the EF-S 18-55mm IS kit lens for Canon APS-C cameras What are the alternatives to Notepad++ on Ubuntu? How to solder with one hand? I am a beginner at electronics.  I don't have many tools and stuff, so I am forced to solder by hand.   I don't have a pcb to solder on, so I usually just solder the wires together, but it's really difficult without a holder.  Is there an easier way to do this? If not, is there a substitute for a wire holder?  Any help would be appreciated. It is impossible for $(x-1)^2+x^2+(x+1)^2$ to be a perfect square Prove that it is impossible for three consecutive squares to sum to another perfect square. I have tried for the three numbers $x-1$, $x$, and $x+1$. Consider these two sentences: "I gave him a pencil," and, "I gave a pencil to him."  Is it correct that the important part of the sentence is placed at the end? When we want to emphasize the pencil that I gave him, must we say, "I gave him a pencil?" When we want to emphasize that it is him to whom I gave a pencil, must we say, "I gave a pencil to him?" Some common problems with displaying routes created from street data, such as bus routes, include:   Routes may overlap themselves, such as when a route takes the same road in opposite directions Routes may overlap other routes that take the same street. Routes are often overly detailed, or "noisy", owing to their derivation from street data. Indicating a route's direction unambiguously without overloading it with arrows can be a challenge.   What are some techniques for dealing with these issues in order to create a practical and usable route map, such as this ? Do you know of any such public transport maps that were produced by automatic techniques rather than manual illustration/digitization?  Specifically I am using ArcGIS 10, but solutions involving other software are welcome as well.  One technique I have had some success with for dealing with 1) is using ArcMap's  with a negative offset -- assuming the lines are oriented in the direction of travel, this will offset the lines such that the right and left-hand sides are distinct and oriented as you would expect (at least for  countries, for left-hand traffic countries one would use a positive offset). However the result tends to be a bit wonky around turns, interchanges and and U-turns, and is inflexible when dealing with multiple routes.  Another possibility that I am not very familiar with but which seems promising is  -- however, I have not had much luck in finding examples of using representations for transportation maps and whether it might fit the bill. This issue happens only on UEFI systems with enabled Secure Boot.  When I try to install DKMS modules like VirtualBox, Nvidia, or Broadcom drivers, they do not install and I get Required key not available when I try to modprobe them.  VirtualBox complains that vboxdrv is not loaded.  Broadcom wl driver is shown in lspci -k as a kernel module but is not in use. sudo modprobe wl throws Required key not available.  Also this issue may happen when I install some kernel modules from git sources.  This issue may appear after a kernel update as disabled wireless adapter, black screen after a reboot, etc.   How can I fix it? Calculate distance of galaxy 13 billion years ago How can I calculate the distance of a point from Earth now vs 13 billion years ago? The excerpt below claims it is 8 times further now. Can someone provide relevant equations ? Smoothing/generalizing polygon in QGIS? Is it possible to smooth or generalize a polygon in QGIS, perhaps by using a plugin like the generalizer-plugin for lines?  I could convert my polygons to lines, smooth them and convert them back to polygons but this is not working for my polygons because of several reasons.