Find all positive integers such that x^y=y^x. Given the graph of x^a and a^x intecepts once at a, it will intercept again and large x as a^x dominates for large x(how could I prove this). Also playing around I have found the solution for x^a=a^a, is a and r^(r/r-1), but this doesn't really answer the question does it as I have defined x=ar, and for a fixed a I can't find x except the obvious solution.
We know that $2^4 = 4^2$ and $(-2)^{-4} = (-4)^{-2}$. Is there another pair of integers $x, y$ ($x\neq y$) which satisfies the equality $x^y = y^x$?
How do I install an SSH private key generated by PuTTYgen? Better way of saying "those who are curious"? I am writing a note in which I want to ask (curious) readers to refer another paper. The phrase curious minded comes to mind, but it seems somewhat awkward.  Is there a better way to convey similar import? Is it OK for OSB3 boards to get wet during construction? How can you find and replace text in a file using the Windows command-line environment? For which $a \in \mathbb{C} -\ \{0,1\}$ does the equation $f(z) = f(az) $ has a non constant solution $f$ with $f$ being analytical in a neighborhood of $0$.  My attempt:  First, we can see that any such solution must satisfy:  $f(z)=f(a^kz)$ for all $k \in \mathbb{N} $.  If $|a|&lt;1$:  The series $z_{k} = a^k$ converges to 0 which is an accumulation point, and $f(z_i)=f(z_j)$ for all $i, j\in \mathbb{N} $. Thus $f$ must be constant.  If $|a|=1$:  For all $a \neq 1$ , $f$ must be constant on any circle around $0$, so again $f$ must be constant.  My quesions are:   Am I correct with my consclusions?   Also, I'm stuck in the case where $|a|&gt;1$. Any ideas?  Thanks Character Limit in HTML How to compute the  formula $\sum  \limits_{r=1}^d r \cdot 2^r$? Given $$1\cdot 2^1 + 2\cdot 2^2 + 3\cdot 2^3 + 4\cdot 2^4 + \cdots + d \cdot 2^d = \sum_{r=1}^d r \cdot 2^r,$$ how can we infer to the following solution? $$2 (d-1) \cdot 2^d + 2. $$   Thank you What the pointer size in 64 bits computer in C++? How do I export the code documentation in C# / VisualStudio 2008? I have always made a point of writing nice code comments for classes and methods with the C# xml syntax. I always expected to easily be able to export them later on.  Today I actually have to do so, but am having trouble finding out how. Is there something I'm missing? I want to go Menu->Build->Build Code Documentation, but there is no option to do that, there. How to convert local time string to UTC? Can I create a TIN from contours lines (shp) in QGIS? I need to do a slope analysis, and would like to make a TIN from 2m contours lines.  Or do a slope analysis directly from contours? The only DEM available is not precise enough.  I do not have any elevation raster files.  And how do I then make slope classes?  The TIN properties does not seem to be able to classify anything other than elevation. Why does one get an illusion as moon following him? When you run or ride bike at night if you observe the moon you feel like he moves along with you as the same speed you are going. Why? How can I assign the output of a command to a shell variable? I want to assign the result of an expression (i.e., the output from a command) to a variable and then manipulate it – for example, concatenate it with a string, then echo it.  Here's what I've got: #!/bin/bash cd ~/Desktop; thefile= ls -t -U | grep -m 1 &quot;Screen Shot&quot;; echo &quot;Most recent screenshot is: &quot;$thefile;  But that outputs: Screen Shot 2011-07-03 at 1.55.43 PM.png Most recent screenshot is:   So, it looks like that isn't getting assigned to $thefile, and is being printed as it's executed. What am I missing? Why doesn't matter pass through other matter if atoms are 99.999% empty space? The ghostly passage of one body through another is obviously out of the question if the continuum assumption were valid, but we know that at the micro, nano, pico levels (and beyond) this is not even remotely the case. My understanding is that the volume of the average atom actually occupied by matter is a vanishingly small fraction of the atom's volume as a whole. If this is the case, why can't matter simply pass through other matter? Are the atom's electrons so nearly omnipresent that they can simultaneously prevent collisions/intersections from all possible directions? prevent php classes from being extends I dropped my camera while on holiday last summer. Thankfully, my camera wasn't damaged but the kit lens has lost the ability to autofocus and the UV filter and 58mm to 52mm filter step-down (I use it to attach a fisheye filter to the lens) I had on its end are stuck and won't come off.  I've replaced the lens with an 18-200 zoom lens so I don't especially need it. I was wondering if it was worth getting the lens looked at and possibly repaired?  I've heard many places will charge you £50 or so just to look at the lens and given the cost of the kit lens I'm not sure it's worth it. Some faces of a model don't respond to UV Mapping I can't seem to get these particular faces to take the UV Mapping. I selected the entire object, entered edit mode, hit U>Unwrap.   The rest of the mesh takes the uv wrapping, but not those faces for some reason.    It looks normal in solid mode.     Does anyone know what could be causing this? Factoring $-3x^2 - 2xy + 8y^2 + 11x - 8y - 6$ How can I extract a predetermined range of lines from a text file on Unix? What properties do natural isomorphisms between functors preserve? If $F,G: C \rightarrow D$ are functors between regular categories and $F \Rightarrow G$ is a natural isomorphism, is it true that if $F$ is faithful {resp. full) then $G$ is full?  I believe this is true but would like some confirmation.  What about if $F$ is essentially surjective?  Is it necessarily true that $G$ is essentially surjective?  Thanks.