Question:  Suppose $ \frac{1}{x^2}+x^2$ is an integer. Prove that $ \frac{1}{x^{2n}}+x^{2n}$ is an integer for all natural $n$.  Hint: Use Strong Induction   My attempt:  Base Case is trivial.  I.H: Assume the result is true for $n = 1,2, ...., k.$  Consider $n = k+1$.  $ \frac{1}{x^{2\left(k+1\right)}}+x^{2\left(k+1\right)}\ =\ \frac{1}{x^{\left(2k+2\right)}}+x^{2k+2}$.  I am not sure what to do from here and how to use the induction hypothesis.
I am to prove by induction that given that $a+1/a$ is an integer (i.e. belongs to Z  ) then $a^n+1/a^n$ is an integer too. I'm pretty much clueless here. Thanks in advance.
Let $a$ be a non-zero real number. Is it true that the value of $$\int\limits_0^\infty \frac{\mathrm{d} x}{(1+x^2)(1+x^a)}$$  is independent on $a$? I've installed ubuntu 17.04 and I want to upgrade it to 18.04. When I tried to do:   sudo do-release-upgrade   I got a message saying An upgrade from 'zesty' to 'bionic' is not supported with this tool. Need inputs to proceed on this.  sudo do-release-upgrade Checking for a new Ubuntu release Your Ubuntu release is not supported anymore. For upgrade information, please visit: http://www.ubuntu.com/releaseendoflife  Get:1 Upgrade tool signature [819 B]                                            Get:2 Upgrade tool [1,257 kB]                                                   Fetched 1,258 kB in 0s (0 B/s)                                                  authenticate 'bionic.tar.gz' against 'bionic.tar.gz.gpg'  extracting 'bionic.tar.gz'  Reading cache  Checking package manager  Can not upgrade   An upgrade from 'zesty' to 'bionic' is not supported with this tool. Why do we use "have" with does and not "has"? I can't really remember how long ago it was, but I think the book was marketed towards children/teens.  All I remember is the basic premise of the story.  Some story about these unusual little blue electric critters that only certain people could see, and these kids took it upon themselves to go about destroying them since they fed off energy or something like that.  Not sure whether they fed off people or off electricity... or both maybe?  I think the dominant method of destroying them was an electric baton that was taken from law enforcement.  I wanna call them buzz batons, but that's probably thanks to Artemis Fowl. Can someone decide to be hit? History substitution fails when implemented in shell script The below command works in command line  rsync -avh -r /Source/09_03_2016/ /Destination/ echo $(basename !:3)      Output 09_03_2016   But when I do this in shell script it does not work  #! /bin/bash  /usr/bin/rsync -avh -r /Source/09_03_2016 /Destination/; echo $(basename !:3)      Output  !:3 How are large amounts of money expected to be made during the early parts of the Waterdeep: Dragon Heist adventure? My public website name and AD domain name are the same. How can I get to my external website from inside my network? I'm running Fedora 25 and I need to take a few particular screen captures so I've read that shutter is a great tool for that. Sadly I am unable to use it for some reason. Whenever I try to take a screenshot this is the result: I attempted a proof of mathematical induction using an arbitrary base case, but was unsuccessful (and hence this question). Below is what I was trying to do and along with my thinking; if anyone can point me in the right direction I'd appreciate it.  The induction I am using: Let $P$ be a property about the natural numbers $\mathbb{N}$ with $0\in\mathbb{N}$, and let $P(n)$ denote the statement that the property $P$ holds for $n\in\mathbb{N}$. Suppose $P(0)$. Furthermore suppose that for each natural number $k$, $P(k)$ implies $P(k+1)$. Then $\forall nP(n)$.  What I am trying to prove: Let $P$ be a property about the natural numbers $\mathbb{N}$ with $0\in\mathbb{N}$, and let $P(n)$ denote the statement that the property $P$ holds for $n\in\mathbb{N}$. Suppose for $n_0\in\mathbb{N}$, $P(n_0)$. Furthermore suppose that for each natural number $k\geq n_0$, $P(k)$ implies $P(k+1)$. Then $\forall n\geq n_0 P(n)$.  My attempted proof: define $Q(n)$ to be $n\geq n_0 \to P(n)$. Then we wish to prove $\forall nQ(n)$. We induct on $n$.  Base case (for ordinary induction): $Q(0)$ is $0\geq n_0\to P(0)$. Since $n_0\in\mathbb{N}$, $0\geq n_0$ implies that $n_0=0$. Since $P(n_0)$, $P(0)$, which proves the base case.  Inductive step: we want to show $\forall n(Q(n)\to Q(n+1))$. To do this, we assume $k\geq n_0 \to P(k)$ and try to show $k+1\geq n_0 \to P(k+1)$.  Since $k\geq n_0 \to P(k)$, we first prove the case where $k\geq n_0$ and $P(k)$. By the hypothesis of the proof, we see that $P(k+1)$, which proves the case for $k+1$.  This is where I am having trouble: For $k&lt;n_0$, I am unable to show the implication for $k+1$.  So my questions would be: (1) is the overall approach for the proof correct? (2) If so, how might I go on to prove the case when $k&lt;n_0$?  Thanks in advance. (This is not homework, by the way.) Can I recover app data after the phone has been Factory Reset? Without remembering application data, I reset my Galaxy S3 to factory state. Now, I lost data for one expense manager app. Is there any way to recover the data? Who would get the longest road? Imagine the following setup. There are 3 (or more) players with an equally long road. What would happen if the longest road was broken? How is it decided who obtains the longest road from there on? C# difference between == and Equals() Right now, I have everything in a partition (except EFI system partition). How can I split the partition into two, one for / and one for /home?   How would you use parted to do that step by step?  How would you use gparted to do that step by step?  How would you use LVM to do that step by step?   One thing I am not sure of is that I can't just shrink the partition and create a second partition on the space that is freed by shrinking, because that doesn't move /home to the new partition.  Thanks.    $ sudo parted -l [sudo] password for t:  Model: ATA TOSHIBA MQ01ABF0 (scsi) Disk /dev/sda: 500GB Sector size (logical/physical): 512B/512B Partition Table: gpt Disk Flags:   Number  Start   End    Size   File system  Name                  Flags  1      1049kB  538MB  537MB  fat32        EFI System Partition  boot, esp  2      538MB   500GB  500GB  ext4  $ sudo df -h [sudo] password for t:  Filesystem      Size  Used Avail Use% Mounted on udev            1.7G     0  1.7G   0% /dev tmpfs           340M  1.4M  338M   1% /run /dev/sda2       457G  6.7G  428G   2% / tmpfs           1.7G   27M  1.7G   2% /dev/shm tmpfs           5.0M  4.0K  5.0M   1% /run/lock tmpfs           1.7G     0  1.7G   0% /sys/fs/cgroup /dev/sda1       511M  6.1M  505M   2% /boot/efi tmpfs           340M   20K  340M   1% /run/user/1000 tmpfs           340M     0  340M   0% /run/user/106 Can I be automatically notified of prolonged blocking in SQL server? About once a week I have to resolve a blocking chain on a SQL Server 2005 database, caused by a long-lived read lock from an Access 2003 front-end. The lock is taken out whenever a user opens a certain form and is released once the user has finished scrolling through the form or closes it. Since many of our users open this form as reference, these locks stay around for a while. Any update to the table causes the blocking, and suddenly nobody can select from this table since they're all waiting on the first lock. This is quite a problem for us, since lots of apps rely on this data. I understand that this locking behavior is part of how Access works with linked tables.  I've been solving the problem from Activity Monitor,  whenever I find out about it. This is a problem not only because it takes me time to do it manually, but also because it's reactive. By the time I hear about it, it's already been a problem for a lot of people.  I'd like to know if there is an automatic way to check for these long-lasting blocking chains, and either be emailed or have the problem resolved automatically. The logic seems straightforward enough ("if any process matching this SELECT query has been blocking for longer than a minute, notify me/kill it") but I don't know how to implement this with SQL Server.  For what it's worth, I think the proper solution is to fix or rewrite the app. However, due to departmental politics, this is not an option for the next few months, so I'm looking for a stopgap. What is the relationship between size of object with distance? How does the size/length of an object vary with distance?   Is it a logarithmic relationship? exponential? linear?   I plotted a curve of the size/length of an object for different distances from the camera, and the curve looked exponential/logarithmic. I was trying to understand the reasoning behind that. How do we see? Where do the photons disappear? I know that the light is reflected from a object to my eyes, but I don't understand exactly how. The photons appear from the light source and disappear in my eye! Can someone explain the phenomenon of where the photons go and do to allow us to see? Let $R=\frac{\mathbb{C}[x,y]}{\langle x^2-y^2-1 \rangle}$. Prove that (1) $R$ is an integral domain. (2) Any nonzero prime ideal of $R$ is maximal.  My idea: (1) The first part is easy. Since $x^2-y^2-1\in \mathbb{C}[y][x]$ is Eisenstein with respect to the prime $y+i$, it is irreducible. Hence the ideal generated by it is a prime ideal. (2) I am struggling with this part. I have one approach (not sure if correct). Approach : Let $I=\langle x^2-y^2-1 \rangle$. Then $R=\mathbb{C}[x,y]/I$.  Any ideal of $R$ is of the form $J/I$ where $J\subseteq \mathbb{C}[x,y]$ is an ideal containing $I$. If we can show that any prime ideal $J$ containing $I$ is maximal, then we are done. How to prove this? Is there any other way to prove the second part? Edit: I am trying to avoid Hilbert Nullstellensatz and the notion of dimension of a ring, as these concepts are not there in the syllabus of the exam from which I found this question. Warning  If you're looking for advice how to make your document look like it's been written in Word, this is most likely not the question you're looking for. This question is mostly of theoretical nature, as it results in tiny differences, which will most likely not be noticed by someone who doesn't allow the use of LaTeX.  Questions that might be more helpful for this matter are:           Question  I learned that MS Word uses a slightly different version of the unit "point" (pt) than TeX does:     The 12 point of Word will be PostScript point, which in TeX would be called 12bp. A TeX pt is slightly smaller: it's 1/72.27 inch, while a bp/PostScript point is 1/72 inch. See also    ( in )  I'm writing a paper that would usually be expected to be "typeset" in MS Word, thus I want to use the same font size as Word would.  How do I set a document e.g. in the "12pt" font size that MS Word would use?  In case it matters, I'm using the article document class, Latin Modern (lmodern) as a font with the T1 font encoding and compile with pdfLaTeX, but input on different set-ups is more than welcome. How can I find color codes on Mac? I used to find color codes which I want to use for texts or background colors on htmls with color edit tool of MS paint which pre-installed on Windows. But now I use Mac instead of Windows. How can I find color codes with Mac? Is an software like MS paint pre-installed on Mac?