Let $n$ be an odd number. Prove that $n$ is a prime number $ \iff$ $\frac{T_n(x)}{x}$ is irreducible on $Q[x]$ We have that  $ $ $T_n(x)$ is Chebyshev polynominal please help me :((
Let $T(n,x)$ be the nth Chebyshev polynomial of the first kind and let $U(n-1,x)$ the $(n-1)$th Chebyshev polynomial of the second kind. Would any one kindly help show that   1) $n$ is prime iff $T(n,x)$ is irreducible in $\mathbb{Z}[x]$.  2) $n$ is prime iff $U(n-1,x)$, expressed in powers of $(x^2-1)$, is irreducible in $\mathbb{Z}[x]$.   Many Thanks!!    No "ordering" is implied here. The wording of the question was done in similar fashion  as any question in any math/research question. I do not see how a person who is asking for help would be ordering people to help.   Any way back to the topic,  For the first part of the question, I noticed that if n is prime, then T(n,x) satisfies  Eisenstein's Irreducibility Criterion. But I am not sure how to show if T(n,x) is irreducible then n is prime.
How do i test capacitors? I've just gotten through replacing capacitors on a trio of dead LCD screens (nothing's blown up yet, so far) - they either had one or two capacitors on their inverter circuit SLIGHTLY bloated, and not quite leaky. I ended up replacing all capacitors of the same brand/'colour', even the ones that looked fine, in case.  Now, checking a bad resistor is simple - i can use a standard multimeter to test it, and i tend to check my solders with the continuity testing option of the multimeter.  How would i test a capacitor ? Is there some standard, common way to test one? Seems that I just proved $2=4$. Solving $x^{x^{x^{.^{.^.}}}}=2\Rightarrow x^2=2\Rightarrow x=\sqrt 2$.  Solving $x^{x^{x^{.^{.^.}}}}=4\Rightarrow x^4=4\Rightarrow x=\sqrt 2$.  Therefore, $\sqrt 2^{\sqrt 2^{\sqrt 2^{.^{.^.}}}}=2$ and $\sqrt 2^{\sqrt 2^{\sqrt 2^{.^{.^.}}}}=4\Rightarrow\bf{2=4}$.  What's happening!? Scale-dependent displaying of a layer using OpenLayers plugin? When I am creating a layer in QGIS the layer position/lines does not stay the same when I zoom in closer than, it seems like 1: 1000. I am adding 2 images to explain the problem. The moment I zoom in closer than 1: 1 000 then the original layer does not display correctly anymore.  Please, can you assist in this matter? How to Select Columns in Editors (Atom,Notepad++, Kate, VIM, Sublime, Textpad,etc) and IDEs (NetBeans, IntelliJ IDEA, Eclipse, Visual Studio, etc) Where are some good places to find royalty free stock images? I'm always on the hunt for free stock images to use while I design, what are some good resources for images that are under a a royalty free or creative commons licence? Why are images formed? How do rays of light getting focused at a point create an image?  How are the information of object's shape and dimensions expressed as an image when light rays converge?  Is it understandable by solving wave equation for light? How can I make Ubuntu 18.04 / 18.10 desktop use Unity (be like Ubuntu 14.04)? Is there a command to delete all the files in a directory that haven't been modified in N days?  I need to clean up some old logs. What is that "1 view" which automatically appears after submitting a question? Are the questions submitted on SO and other SE sites initialized with 1 view, or does the first view belong to the user who posted it?  Why is a question given a free first view? It just appears, as soon as the question is published. I need to compute the improper integral $$ \int_{-\infty}^{\infty}\frac{x^{2}}{\cosh\left(x\right)}\,{\rm d}x $$ using contour integration and possibly principal values.  Trying to approach this as I would normally approach evaluating an improper integral using contour integration doesn't work here, and doesn't really give me any clues as to how I should do it. This normal approach is namely evaluating the contour integral $$ \oint_{C}{\frac{z^2}{\cosh\left(z\right)}\mathrm{d}z} $$ using a semicircle in the upper-half plane centered at the origin, but the semicircular part of this contour integral does not vanish since $\cosh\left(z\right)$ has period $2\pi\mathrm{i}$ and there are infinitely-many poles of the integrand along the imaginary axis given by $-\pi\mathrm{i}/2 + 2n\pi\mathrm{i}$ and $\pi\mathrm{i}/2 + 2n\pi\mathrm{i}$ for $n \in \mathbb{Z}$. The residues of the integrand at these simple poles are $-\frac{1}{4}\mathrm{i}\pi^{2}\left(1 - 4n\right)^{2}$ and $\frac{1}{4}\mathrm{i}\left(4\pi n + \pi\right)^{2}$, so that even when we add up all of the poles, we have the sum $4\pi^{2}\mathrm{i}\sum_{n = 0}^{\infty}\,n$, which clearly diverges.  Any hints would be greatly appreciated. To find minimum distance between two curves Does $\frac{n}{\sum\limits_{k=1}^{n}\Big(\frac{k}{k+1}\Big)^k}$ converge? Does the sequence $$\displaystyle \frac{n}{\sum\limits_{k=1}^{n}\Big(\frac{k}{k+1}\Big)^k}$$   converge?   Attempt. Since $\Big(\frac{k}{k+1}\Big)^k \rightarrow 1/e\neq 0$ and the terms are positive, the series $\sum\limits_{k=1}^{\infty}\Big(\frac{k}{k+1}\Big)^k$ diverges to $+\infty$. I find hard to determine if $n$ or the sum goes faster to $+\infty.$   Thanks in advance. Keyboard Glyphs How are people showing keyboard &quot;glyphs&quot; on Super User? I looked at the Markdown help page and it didn't show anything regarding this. Here is  (question has been deleted since this was asked, so an archive link is provided instead): Proof of linear independence of $e^{at}$ Given $\left\{ a_{i}\right\} _{i=0}^{n}\subset\mathbb{R}$ which are distinct, show that $\left\{ e^{a_{i}t}\right\} \subset C^{0}\left(\mathbb{R},\mathbb{R}\right)$,  form a linearly independent set of functions.   Any tips on how to go about this proof. I tried a working from the definition of an exponential and combining sums but that didn't seem to get me anywhere. I saw a tip on the internet that said write it in the form  $\mu_{1}e^{a_{1}t}+\dots+\mu_{n}e^{a_{n}t}=0$ to try to show $\mu_{1}=\dots=\mu_{n}=0$ considering each term of the left hand side must be positive, but I can't get my head around that because while I understand $e^{x}&gt;0\forall x\in\mathbb{R}$ I cannot see why $\mu_{i}$ must be positive in any case. I have thought about differentiating but that doesn't seem to help. The question did originally ask for a "rigourous" proof but I'll take any hints right now and the provided the solution of 'is obvious' is most unhelpful to me.  Any input would be fantastic. Thank you. My recollection is very sketchy but this is what I remember....   Film was possibly black and white I saw it as a child in the late '80s early '90s but it's older It was subtitled and in Russian or similar language It involved some astronauts  travelling in space with possibly time travel The astronauts (mostly men) encounter a planet/civilization of only women The women no longer need men to breed and are so become superior and possibly want to eliminate all men    That's all I have to go on, sorry. Confused about the possibility of different differentiable structures. In Loring Tu's "An Introduction to Manifolds" an atlas on a manifold is a collection of coordinate charts that are pairwise compatible and cover the manifold. A smooth manifold is defined to be a topological manifold together with a maximal atlas or differentiable structure. The maximal atlas is constructed by taking an atlas and appending all coordinate charts that are compatible with the atlas and using this he shows that any atlas on a locally Euclidean space is contained in a unique maximal atlas.  He also later shows that for any chart on a manifold, the coordinate map is a diffeomorphism onto it's image.   My question arises when Tu mentions in an aside that every compact topological manifold in dimension four or higher has a finite number of differentiable structures. What I don't understand is how it is possible for a smooth manifold to have more than one differentiable structure. Doesn't the fact that for any chart the coordinate map being a diffeomorphism coupled with the fact that composition of smooth functions is smooth mean that any two coordinate charts on a manifold are compatible and thus they must all belong to one maximal atlas? How is it possible to have another? Am I not understanding the word "maximal" or atlas or some other concept? Limit question on independent random variables (Exercise 4.2.4 from Grimmett and Stirzaker) Let $\{X_r | r \geq 1\}$ be independent and identically distributed with distribution function $F$ satisfying $F(y) &lt; 1$ for all $y$, and let $Y(y) = \min \{k | X_k &gt; y\}$. Show that $$ \lim_{y\rightarrow \infty} P(Y(y) \leq E[Y(y)]) = 1 - e^{-1}. $$  I managed to show that $E[Y(y)] = \alpha(y) = (1 - F(y))^{-1}$ and that $$ \lim_{y\rightarrow \infty} P(Y(y) \leq E[Y(y)]) = \lim_{y \rightarrow \infty} \left( \sum_{i = 1}^{\lfloor \alpha(y) \rfloor} {\lfloor \alpha(y) \rfloor \choose i} (-1)^{i+1} \alpha(y)^{-i} \right). $$ And the expression on the right, according to Mathematica, does indeed seem to converge to $1 - e^{-1}$. But how can I show that this is true? Can anyone give me a hint? Beamer and description environment text size Can store bought salmon be used for sashimi? Is it safe to consume raw fish purchased at the fish counter of a typical suburban grocery store? If not, do you have any tips for finding sushi grade fish. Also, are there requirements for safe sushi/sashimi beyond the use of safe fish? How to increase battery life on Ubuntu 20.04 and what power saving software should I install, if any?