is there a formula for the sum of n'th powers
Sum of $k$-th powers
Let $F$ be a field and $F[x]$ a polynomial ring. Let $p(x)$ be an irreducible polynomial.  Show that $\gcd(p(x),q(x)) = 1\Longrightarrow \exists r(x),s(x)$ such that $r(x)p(x)+s(x)q(x) = 1$.  I know the proof for Bezout's identity for , but this proof uses the notion of absolute value, which cannot be applied to a polynomial ring.   I am thinking about a way to prove the statement: Use the Euclidean algorithm to compute $\gcd(p(x),q(x))$ and show that the result is $1$, and somehow go backwards to obtain $r(x)p(x)+s(x)q(x) = 1$, but it is a bit messy, and I'm not yet able to see it completely.  Please offer some help. If you have some other methods, that would be great, too. I was working with a new C++ developer a while back when he asked the question: "Why can't variable names start with numbers?"  I couldn't come up with an answer except that some numbers can have text in them (123456L, 123456U) and that wouldn't be possible if the compilers were thinking everything with some amount of alpha characters was a variable name.  Was that the right answer?  Are there any more reasons?  string 2BeOrNot2Be = "that is the question"; // Why won't this compile? Accessing private member variables from prototype-defined functions Is there any way to make “private” variables (those defined in the constructor), available to prototype-defined methods?  TestClass = function(){     var privateField = "hello";     this.nonProtoHello = function(){alert(privateField)}; }; TestClass.prototype.prototypeHello = function(){alert(privateField)};   This works:  t.nonProtoHello()   But this doesn’t:  t.prototypeHello()   I’m used to defining my methods inside the constructor, but am moving away from that for a couple reasons. Number of finite simple groups of given order is at most $2$ - is a classification-free proof possible? Is it possible to restore a MySql database only using files from Data directory? I have a mysql installed inside a windows virtual machine. The hard drive was crashed and the system is not able to restore all again. But I can have access to entire c: ROOT. So I went to  c:/ProgramData/Mysql/Mysql Server 5.5/data/myDatabaseName    There I can see different files like: .frm, .opt, .trg and .trn. In a regular way, I could run a mysqldump to backup and restore all database from command line, but the system is not able to be restored. So I had to copy all these files and past inside inside the same  directory inside same database in another physical instance.   The issue is some tables are not been recognized by this new instance. So I think copy and past is not enough to restore the database.   What am I missing?    Note: Database has innoDb and Myisam tables. On my system (Windows Vista Home Premium 64bit) Python files are associated with Python interpreter:  C:\Users\Piotr&gt;assoc .py .py=Python.File C:\Users\Piotr&gt;ftype Python.File Python.File=c:\python\2.7\python.exe "%1" %*   Nevertheless when I run any Python script from command line a window titled Open With pops up asking me to Choose the program you want to use to open this file. Validate/verify PDF files integrity There is a length limit to Windows file extensions? I was trying to fix my audio problems using suggestions from . One of the comments suggested I execute this and only this:     killall pulseaudio   So I did, and I restarted my computer just in case, but all it did was remove my soundcard and I still couldn't hear anything. Now I don't know how to get my soundcard back.  Please note that I am extremely inexperienced when it comes to Ubuntu. Update help center article on search to reflect the new position of the search box The page  says:      Enter search terms in the search box that appears in the upper right corner of every page    But the box is not there. better Curve/Text to Mesh generation? UEFI machine doesn't boot Ubuntu through NVRAM bootcatalog. How to fix? BibTeX: Abbreviate first name (aka given name) to 2 or 3 letters (not 1) Single word that means "to look down on others"? How to display javascript console on Chrome (or a different browser)? I want to quickly try some Javascript code on Android phone.   I cannot find how to open console or something similar on either Chrome or default browser. about:debug doesn't do anything, and I don't want "remote debugging".   I can download another app if necessary (but I don't want to root my device). Best way to start investing, for a young person just starting their career? I am young, just started my career, and looking to begin investing so I can make my money work for me. I'm still living like I did in college (occasional purchase on myself, but most of it goes into savings) and I would like to take the extra money and invest it.   What would you suggest are the best options for me to begin? I was considering getting a personal investor [a financial advisor] for perhaps a year or so, but I've heard that they really are not worth the money. I wanted to get the input of the great minds here. Intuituive reason why Fermats last theorem holds I am unsure of whether it is normal, but to me, intuitively Fermats last theorem should not hold.  If anyone intuitively believed it to be correct, why?     Can someone explain so I understand somewhat why FLT holds without reading a massive and complex proof? I followed Mike's solution from my first question:    My next question is how to add entries (for the abstract and acknowledgements pages) into the table of contents. I am trying to create a 3d game in unity. I have a cube in unity and I want it to move left or right on swipe . It should move to a distance till where user swipe or take their finger off of a screen. I want it to move fast or the cube should move fast on swipe and cube should flow smooth. Like in the gif there is a circle and the user use their finger to move the circle till wherever the user swipe it and it should be instant and I want my cube to move like that at a distance till where user swipe their finger and it should be instant and fast.  I don't know how to do it. Any help is appreciated! How do you prove that $\sum\limits_{n=0}^{\infty}{2n \choose n}x^n=(1-4x)^{-1/2}$? I tried to identify the sum as a binomial series, but the $4$ and the $-1/2$ puzzle me. (This series arises in studying the first passage time of a simple random walk.)