My teacher asked me this question. But I  think this is wrong. Anyone found this, before me? I do not know. Anyway, Is my solution correct?  ${x^{x^{x^{x^x}}}}^{...}=2$  $x^2=2$  $x=\sqrt 2$  ${\sqrt2^{\sqrt2^{\sqrt2^{\sqrt2^\sqrt2}}}}^{...}=2$  Now,  Let, ${x^{x^{x^{x^x}}}}^{...}=4$  $x^4=4$  $x=\sqrt2$  ${\sqrt2^{\sqrt2^{\sqrt2^{\sqrt2^\sqrt2}}}}^{...}=4$  Contradiction!
Find $x$ in   $$ \Large 2 = x^{x^{x^{\:\cdot^{\:\cdot^{\:\cdot}}}}}$$   A trick to solve this is to see that $$\large 2 = x^{x^{x^{\:\cdot^{\:\cdot^{\:\cdot}}}}} \quad\implies\quad 2 = x^{\Big(x^{x^{x^{\:\cdot^{\:\cdot^{\:\cdot}}}}}\Big)} = x^2 \quad\implies\quad x = \pm \sqrt{2} $$  Are these solutions correct? If not, why? If yes, are there other solutions?    PS: An extension of this discussion can be found in
Generator of rational functions unchanged under $\sigma(X) = X + 1$ Let $L = K(X)$ be the field of rational functions over a field $K$ with characteristic $p &gt; 0$, and let $\sigma \in \operatorname{Aut}_K(L)$ with $\sigma(X) = X + 1$.   Show that $G = \left&lt;\sigma\right&gt;$ is finite, and determine a generator of $L^G = \{ f \in L: \rho(f) = f$ for all $\rho \in G \}$ over $K$.    My solution so far: $G = \left&lt;\sigma\right&gt;$ is finite because char $(K) = p &gt; 0$ and therefore $\sigma^p (X) = X + p = X $.  A rational function unchanged under $\sigma$ will also be unchanged under any $\sigma^n$, so we don't have to worry about powers of $\sigma$ I think.   Any ideas for finding a generator? Thanks. I have written a small script to count number of occurrences of logs in a particular file. 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They work correct with the setView function.  var map = L.map('leaflet_map').setView([52.23943, 4.97599], 15);   But using a geojasonFeature they are, hmmm, 'projected', somewhere east of Somalia.  var geojsonFeature = {             "type": "Feature",             "properties": {             "name": "Coors Field",             "amenity": "Baseball Stadium",             "popupContent": "This is where the Rockies play!"             },             "geometry": {                 "type": "Point",                 "coordinates": [52.23943, 4.97599]             }         };         var myLayer = L.geoJson().addTo(map);         myLayer.addData(geojsonFeature).bindPopup("I am a gjson point.");   Anyone who knows what is happening here?  EDIT  Out of pure curiosity I changed the coordinates around [4.976143930893815,52.23925499011473] and the point appears at its correct location. A known bug!? Access Java / Servlet / JSP / JSTL / EL variables in JavaScript Factorial Inequality problem $\left(\frac n2\right)^n > n! > \left(\frac n3\right)^n$ I met an inequality, I ask, do not mathematical induction to prove that:     Prove \[ \left(\frac n2\right)^n > n! > \left(\frac n3\right)^n \] without using induction Multiple TikZ keys with foreach top output: cpu usage > 100% Here is the top output which I gathered:    I noticed that top is showing vlc's cpu usage is > 100%.  Can anybody please justify me why it is showing like this, whether this is a bug in top application or something else. How to change SQL Server collation How many questions on meta have asked, "Why can't I ask this question when that question is open"? When performing machine learning, the performance of the machine learning method is dependent on the original function $f$ that we are trying to learn (let's forget for a moment the non-deterministic error and focus on the properties of $f$).  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We can note that the sequence $\lbrace S_n \rbrace$ is a strictly increasing sequence, with $\pi^2/6$ as its lowest upper bound, so I thought that maybe the sequences $\lbrace A_n \rbrace$ and $\lbrace B_n \rbrace$ both have no upper bound, and the sequences would tend to infinity, and if I were able to prove this, I thought I could use it to argue that $S$ is an irrational number, using the fact that every rational number can be written as the quotient between two finite integers.  We note that $A_n\geq B_n$ for all natural $n$, as $S_n\geq 1$, so all we would really need to prove in that case is that $\lbrace B_n \rbrace \rightarrow \infty$(if this assertion is even correct).  At first I thought that both $\lbrace A_n \rbrace$ and $\lbrace B_n \rbrace$ would be increasing sequences, but after checking with Maple I noticed that they weren't, sadly enough($S_9=\frac{9778141}{6350400}$ and $S_{10}=\frac{1968329}{1270080}$). 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