I am interested in understanding why $x^4 - 17 = 2 y^2$ has no rational solutions.  My first approach was to show that every rational solution leads to an integral solution but that doesn't seem to be the case.
There is a hint to show that if there were points in $\mathbb{Q}$, then there would exist $r,s, t \in \mathbb{Z}$ with $\gcd(r, t) = 1$ such that $2s^2 = t^4 - 17r^4$ , and then show that any prime dividing $s$ is a quadratic residue modulo $17$.  I'm stuck on the first bit - if you suppose $\left(\frac{a}{b}, \frac{c}{d}\right)$ is a point on the curve in $\mathbb{Q}$ (so $a, b, c, d \in \mathbb{Z}$) then you get to $2a^2d^4 = b^2c^4 - 17b^2d^4$, but I'm not sure where to go from here since you don't know that $b$ is a square?
How to close view windows? I would like to close some view windows that I opened accidentally.  See the image below:    Edit: another note: I tried searching for the solution, but only found things like "Join Area", which don't seem to work in this situation. I've been trying to solve the following problem:  Suppose that $f$ and $f&#39;$ are continuous functions on $\mathbb{R}$, and that $\displaystyle\lim_{x\to\infty}f(x)$ and $\displaystyle\lim_{x\to\infty}f&#39;(x)$ exist. Show that $\displaystyle\lim_{x\to\infty}f&#39;(x) = 0$.  I'm not entirely sure what to do. Since there's not a lot of information given, I guess there isn't very much one can do. I tried using the definition of the derivative and showing that it went to $0$ as $x$ went to $\infty$ but that didn't really work out. Now I'm thinking I should assume $\displaystyle\lim_{x\to\infty}f&#39;(x) = L \neq 0$ and try to get a contradiction, but I'm not sure where the contradiction would come from.   Could somebody point me in the right direction (e.g. a certain theorem or property I have to use?) Thanks Prove that $\mathbb{Z}$ is not isomorphic to additive group of any vector space over any field.  Proof.  Surpose that: $\phi : (A, +) \rightarrow  \mathbb{Z} $ is an isomorphism. Then there is some $r∈A $ such that $ϕ(r)=1_\mathbb{Z}$. We have $$ 1_\mathbb{Z}=ϕ(r)=ϕ(2(r/2))=2ϕ(r/2)$$  or equivalently that $ϕ(r/2)=\frac{1}{2}$. But this is not in $ \mathbb{Z}$, so there can be no such morphism.  Is my proof above correct? Thanks Examples of bijective map from $\mathbb{R}^3\rightarrow \mathbb{R}$ Rotating hatch labels based on orientation changes in Data Driven Pages Strip maps? I've created a strip map for a linear feature and have it loaded as a data driven page.  I want to show stationing for the line.  The stationing displays properly on the first data driven page but on each subsequent page the stationing labels get messed up as the map orientation changes. I want the stationing labels perpendicular to the linear feature as shown in the first image.  The tick marks seem to rotate and stay perpendicular to the line but the labels won't.  How can I fix this? I am wondering if when an email is forward does it lose its original headers? Set limit to one author when using "et al." in biblatex Preserving changed Output style Can I mask an input text in a bat file? Why can't Chromium suddenly access any partition except for /home? I’m not sure What it is? I’m trying to get a glossy shade to building in my scene as soon as I put roughness to zero I see these trees in glass. I have no trees in my scene anywhere, background is black. Even when I render I see these trees. How can I get rid of them and still keep my glossy building material. Python scope: "UnboundLocalError: local variable 'c' referenced before assignment" I am trying to figure out this:  c = 1 def f(n):     print c + n  def g(n):     c = c + n  f(1) # =&gt; 2 g(1) # =&gt; UnboundLocalError: local variable 'c' referenced before assignment   Thanks! While we know that every finite set is compact, how to prove that every infinite subset of $\mathbf{R}$, the set of all real numbers, is also compact in the finite-complement topology (also called the co-finite topology)?   What is the situation in the case of an arbitrary infinite set $X$ with the finite-complement topology? What is the plural form of "Software"? Is it correct to use softwares, as in the following sentence?  There are various video conversion softwares in the market. Expected Value of Max of IID Variables What is the expected value of the maximum of 500 IID random variables   with uniform distribution between 0 and 1?   I'm not quite sure of the technique to go about solving something like this. Could anyone point me the right direction?  Thanks Etiquette, rules, or advice on which site to pose a question? (Or, why are there so many StackExchange sites?) I am trying to hide Authoring information, Revision information, Publishing options tabs in create/edit node for any content type. I tried different modules with no luck. Please give me suggestions how to achieve this. Thanks a lot Is there an easy way to 'break' the axis in a pgfplots environment? What I'm talking about is something like     Any minimal plot will do as an example...for example  \documentclass{minimal}   \usepackage{tikz} \usepackage{pgfplots}  \begin{document} \begin{tikzpicture} \begin{axis}[ymin=0, ymax=80] \addplot {x*0};  \addplot {x^2+50}; \end{axis} \end{tikzpicture}  \end{document}   For example, in the below plot,    I want to break the axis from around y = 10 to y = 40.   Any ideas? Does PGF have this functionality? A search for `break' in the manual doesn't yield anything. Prove that two distinct numbers of the form $a^{2^{n}} + 1$ and $a^{2^{m}} + 1$ are relatively prime if $a$ is even and have $\gcd=2$ if $a$ is odd.   My attempt: If $a$ is even, let $a = 2^{s}k$ for some integers $k, s$ Then, $$a^{2^{n}} + 1 = 2^{2^{n}s}\cdot k^{2^n} + 1$$ and $$a^{2^{m}}  + 1 = 2^{2^{m}s}\cdot k^{2^m} + 1$$ To prove that they're relatively prime, we need to show that their gcd = 1. And I was stuck here, how could I prove that gcd of two numbers is $1$?     A hint would be sufficient. Thanks. convexity proof of a function including ln and sums