Find all positive integers $x,y,z$ such that $$x!+y!=z!$$ My Progress: let $x=y=1$ so that you get $2!$ Which is true. How do I compute this algebraically instead of substituting random values?
I have the formula $x! + y! = z!$ and I'm looking for positive integers that make it true. Upon  inspection it seems that x = y = 1 and z = 2 is the only solution. The problem is how to show it.   From the definition of the factorial function we know $x! = x(x-1)(x-2)...(2)(1)$  So we can do something like this:  $$ [x(x-1)(x-2)...(2)(1)] + [y(y-1)(y-2)...(2)(1)] = [z(z-1)(z-2)...(2)(1)]$$   we can then factor all of the common terms out on the LHS.  $$ [...(2)(1)][x(x-1)(x-2)... + y(y-1)(y-2)...] = [z(z-1)(z-2)...(2)(1)]$$ and divide the common terms out of the right hand side  $$[x(x-1)(x-2)...] + [y(y-1)(y-2)...] = [z(z-1)(z-2)...]$$  I'm stuck on how to proceed and how to make a clearer argument that there is only the one solution (if indeed there is only the one solution).   If anybody can provide a hint as to how to proceed I would appreciate it.
Why does an open interval NOT have measure zero? Word for a temporary project name In software development, and I imagine in many other fields, we often make up temporary nicknames for new projects or features. These are not meant to be the real names for the end product and are often funny, playful, or sarcastic in nature. These are also not intended to be "codenames" that hide the nature of the project or feature. Is there a word for such nicknames? Are hand motions required to use the Force? In almost every canon (that is, visual) source, Force users typically wave a hand to invoke the Force to move or manipulate objects, people and thoughts. I get out-of-universe this is a visual cue that the Force is being used, but in-universe is it strictly necessary?  (The only exception I can think of is when Luke is training on Dagobah and is balancing rocks while standing on one hand with Yoda on his foot. He doesn't appear to be waving his hand to move the rocks.)  Is this addressed anywhere in-universe, even in Legends? Does $1.0000000000\cdots 1$ with an infinite number of $0$ in it exist? Characterizing units in polynomial rings I am trying to prove a result, for which I have got one part, but I am not able to get the converse part.  Theorem. Let $R$ be a commutative ring with $1$. Then $f(X)=a_{0}+a_{1}X+a_{2}X^{2} + \cdots + a_{n}X^{n}$ is a unit in $R[X]$ if and only if $a_{0}$ is a unit in $R$ and $a_{1},a_{2},\dots,a_{n}$ are all nilpotent in $R$.  Proof. Suppose $f(X)=a_{0}+a_{1}X+\cdots +a_{n}X^{n}$ is such that $a_{0}$ is a unit in $R$ and $a_{1},a_{2}, \dots,a_{r}$ are all nilpotent in $R$. Since $R$ is commutative, we get that $a_{1}X,a_{2}X^{2},\cdots,a_{n}X^{n}$ are all nilpotent and hence also their sum is nilpotent. Let $z = \sum a_{i}X^{i}$ then $a_{0}^{-1}z$ is nilpotent and so $1+a_{0}^{-1}z$ is a unit. Thus $f(X)=a_{0}+z=a_{0} \cdot (1+a_{0}^{-1}z)$ is a unit since product of two units in $R[X]$ is a unit.   I have not been able to get the converse part and would like to see the proof for the converse part. ASP.NET Call Code Behind Function From Javascript How to use unicode characters in Windows command line? We have a project in Team Foundation Server (TFS) that has a non-English character (š) in it. When trying to script a few build-related things we've stumbled upon a problem - we can't pass the š letter to the command-line tools. The command prompt or what not else messes it up, and the tf.exe utility can't find the specified project.  I've tried different formats for the .bat file (ANSI, UTF-8 with and without ) as well as scripting it in JavaScript (which is Unicode inherently) - but no luck. How do I execute a program and pass it a Unicode command line? Make program first read from pipe, then from keyboard How do I find the address of last non-blank cell in a row (while skipping columns) What would be the right position of "substantially" in the following:     1). Before verb: These optimal values substantially contribute to the success of the methodology.      2). After verb: These optimal values contribute substantially to the success of the methodology.   Here is my issue:  Fact1: When I use Google Ngram, I get , 0 occurrence of option 1.  Fact2: I found an excellent Grammar book that discusses Adverbs, their types and prefered positions. The author defines some rules that I'll summarize as follows:   Connecting adverbs, eg: however, then,..., Position: Beginning of clause. Indefinite frequency, eg: always, usually,..., Position: Mid position (after auxilary verb, before other verbs) sometimes beginning of clause. Focusing adverbs, eg: mainly, only,..., Position: Mid position. Adverb of certainty, eg: clearly, obviously, Position: Mid position. Adverb of completeness, eg: almost, nearly,..., Position: Mid position. Adverb of manner, eg: fast, slowly, nicely,..., Position: End clause, mid and beginning. Adverb of place, eg: upstairs, around, Position: End clause, sometimes beginning. Adverb of time, eg: daily, weekly, ... Position: Mostly end clause, sometimes beginning and mid position.   I think that "substantially" is an adverb of completeness. Therefore, the right position is before the verb "contribute" (option 1). However, Google Ngam indicates that option 1 is not widely used.  What do you think? Any suggestions? Since the equation of mechanics are of second order in time, we know that for $N$ degrees of freedom we have to specify $2N$ initial conditions. One of them is the initial time $t_0$ and the rest of them, $2N-1$ are initial positions and velocity. Any function of these initial condition is a constant of motion, by definition. Also, there should be exactly $2N-1$ algebraically independent constants of motion.  On the other hand, Noether's procedure gives us integrals of motion as a result of variational symmetries of the action. These integrals of motion are also conserved but they are not always $2N-1$ in number. In consequence, we classify the system by their integrability.  So, what is the difference between the  and ? Why do non- have less integrals of motion when they should always have $2N-1$ constants of motion? Show that $$\int_{0}^{\infty} \frac{x - \sin(x)}{x^3} \, dx = \frac{\pi}{4}$$   My attempt is as follows: Let $$f(z) = \frac{z - i e^{iz}}{z^3}$$ and consider the contour on $[\epsilon, R]$ followed by a semicircular arc in the counter clockwise direction, then on $[-R, -\epsilon]$, then the semicircular clockwise contour avoiding the origin. We have, then, that  $$0 = \int_{\Gamma} f(z) dz = \int_{[\epsilon, R]} f(t) dt + \int_{C_R}f(Re^{it})Rie^{it}dt + \int_{[-R, -\epsilon]}f(t)dt + \int_{C_{\epsilon}}f(\epsilon e^{-it})\epsilon i e^{-it} dt$$  Then the first and third integrals ( $I_1$ and $I_3$) combine so that  $$I_1 + I_3 = 2\int_{\epsilon}^R \frac{t - \sin{t}}{t^3}\,dt$$  Further,   $$|I_{C_R}| \leq \int_0^\pi \left|\frac{Re^{it} - ie^{-R\sin{t}}e^{iRcos{t}}}{R^2 e^{2it}} \right|dt \rightarrow 0 \text{ as } R\rightarrow \infty$$  (I've omitted the details, it isn't too bad to bound)  However, I'm having trouble computing the limit  $$\lim_{\epsilon \rightarrow 0}\int_{C_{\epsilon}} f(\epsilon e^{-it})\epsilon i e^{-it} dt$$  No matter which way I look at it, it seems like this limit does not exist. Perhaps I'm seeing something wrong or have I chosen a bad $f(z)$? Everytime I try to sudo something at the terminal, I get a warning message:  alexandre@XPS-15Z $ sudo mount file.iso /mnt/ -o loop [sudo] password for alexandre:  no talloc stackframe at ../source3/param/loadparm.c:4864, leaking memory   It's just annoying to get that all the time. How do I solve that? I'm on Ubuntu 14.04 (upgraded from 13.10) Excel INDIRECT function and conditional formatting - highlighting a row Why haven't I received the "Curious" badge? As you can see in the picture, I have 13 out of 5. But it's been like this for about two months! What's the reason for that? Get steam to realize I have games already installed I uninstalled and re-installed steam. I didn't uninstall my games, and they still exist on my hard drive (I checked).  I have an SSD and a HDD, and to save space I install programs (including steam) to C:\Program Files (x86)\Steam drive (SSD) and games to D:\Games\SteamApps (HDD). How to I let steam know the programs are installed there? Suppose I have a document with multiple include or input statements  \input{fileA} \input{fileB}   etc.   Is there an easy way to generate a single .tex file where \input{fileA} is replaced by the actual content of fileA etc. without copying it manually? Test for finite variance? Is it possible to test for finiteness (or existence) of the variance of a random variable given a sample? As a null, either {the variance exists and is finite} or {the variance does not exist/is infinite} would be acceptable. Philosophically (and computationally), this seems very strange because there should be no difference between a population without finite variance, and one with a very very large variance (say > $10^{400}$), so I am not hopeful this problem can be solved.  One approach that had been suggested to me was via the Central Limit Theorem: assuming the samples are i.i.d., and the population has finite mean, one could check, somehow, whether the sample mean has the right standard error with increasing sample size. I'm not sure I believe this method would work, though. (In particular, I don't see how to make it into a proper test.) Sort ArrayList of custom Objects by property Packages: graphics vs graphicx Different websites suggest using one of those two packages.  \usepackage{graphicx} \usepackage{graphics}   What are differences between them? Which is better?