Prove that the diophantine equation $x^4-2(y^2)=1$ has only 2 solutions. Any hint on how to start and what to do .. I do not have a lot of experience on non linear diophantine equations and do not know how to approximate them. So far I can see that for $y=0$, $x=1$ is one solution.
The one of solution of $x^4 - 2y^2 = -1$ is $x = 1$ and $y = 1$. However, the solution $(1, 1)$  of $x^4 - 2y^2 = 1$ is failed. We know $x = 1$ and $y = 1$ is small integers and we can check by trail method. In case more solutions are existing or not how to check? What are the solutions of $x^4 - 2y^2 = 1$?
Is it standard to say $-i \log(-1)$ is $\pi$? I typed $\pi$ into Wolfram Alpha and in the short list of definitions there appeared  $$ \pi = -i \log(-1)$$  which really bothered me. Multiplying on both sides by $2i$:  $$ 2\pi i = 2 \log(-1) = \log(-1)^2 = \log 1= 0$$  which is clearly false.  I guess my error is $\log 1 = 0$ when $\log$ is complex-valued.  I need to use $1 = e^{2 \pi i}$ instead.  So my question: is it correct for WA to say $\pi = -i \log(-1)$?  Or should be they specifying "which" $-1$ they mean?  Clearly $ -1 = e^{i \pi}$ is the "correct" value of $-1$ here. I'm new in LaTex. I need to draw a graphic like the image and each rectangles of image must have the size shown in Figure. I drwan it by MS power point but I need to do by LaTex. Can you help me? Thank you in advance for your help and collaboration!  Thank you in advance for your help and collaboration! How to recover a deleted file from /data partition? I factory reset my rooted Android M phone but now I realize that there was a file in the /data directory which I need.  Is it possible to create an image (.img or something similar) of the /data partition so that I could run some recovery software for ext4 on that image? I have a little knowledge about ADB. Given  $N\geq30$ i.i.d. $X_n\approx\mathcal{N}(\mu_X,\sigma_X^2)$,  and $\mu_X \approx 0$, looking for:   accurate closed form distribution approximation of $Y_N=\prod\limits_{1}^{N}{X_n}$ asymptotic (?) approximation of same product    This is a special case $\mu_X \approx 0$ of a more . Brace expansion not working in a script for i in {1..40} do     echo $i done   I got  {1..40}   and I would like to have something like  1 2 3 and so on   so I can use the variable i inside a command's parameter. Geometric mean never exceeds arithmetic mean This was a mathematical induction question proposed in a textbook, and I've exhausted multiple approaches (proving RHS - LHS > 0, splitting the fraction, fractional exponents, etc.)     The geometric mean of $n$ positive numbers $a_1, a_2,\ldots,a_n$ is $\sqrt[n]{a_1a_2 \ldots a_n}$ and their arithmetic mean is $\frac{a_1+a_2+\ldots+a_n}{n}$. If $a_1, a_2,\ldots,a_n$ are $n$ positive real numbers, prove by induction that their geometric mean is always smaller than or equal to their arithmetic mean, i.e. $\sqrt[n]{a_1a_2\ldots a_n} \leq \frac{a_1+a_2+\ldots+a_n}{n}$ Do I need a transit visa for Malaysia? I am an Indian passport holder travelling from China to India via Kuala Lumpur, on Air Asia tickets bought separately. The layover is 4 hours and 30 minutes.  Do I need a transit visa? Why is there a magnetic field around a magnet if there are no charges moving? Configure Postfix to use external MX servers for delivery of local mail if user is unknown I have a following setup:   linux box with postfix configured to be responsible for example.com domain domain's MX servers are configured so that mail sent to example.com is sent to google mail servers several user accounts on linux machine exist (same machine also hosts example.com site)   When someone from the outside attempts to send mail to address ending with @example.com, it gets routed to google mail (and there handled appropriately).  When linux machine tries to send mail to outside world, mail is delivered correctly, as reverse dns and spf records are configured correctly, so linux machine is valid mail sender for example.com domain (along with google mail servers).  However, here's the problem. When php application (hosted at linux box) tries to send mail to someuser@example.com (and someuser doesn't exist on linux box), it fails, since it doesn't even consult google mail servers, but postfix smtp locally concludes that "someuser" is unknown.  So, the question is: how do I tell postfix to relay mails sent to @example.com domain to google mail servers (so, to servers specified in MX records), IF and only if a mailbox is not found locally. Is it possible to add a list of hosts that are only specific to a certain user? Perhaps a user-specific hosts file?  This mechanism should also complement the entries in the /etc/hosts file. Creating a unique alphanumeric 10-character string I'm looking to create a simple short-lived reservation system, and I'd like to generate confirmation numbers that are   unique random-looking alphanumeric short-ish, at least much shorter than 32 character-long strings returned by sha1   I'm only looking to have ~500 reservations, so I don't imagine high likelyhood of collissions.  One idea I had is generate an sha1 hash based on a date-time stamp and username, then truncating it to its first 10 characters.  Would something like that be reliably unique enough for the purposes of processing ~500 reservations? Am not doing anything fancy, just trying to update a field on Contact in JS of LWC but getting this exception.  Uncaught TypeError: 'set' on proxy: trap returned falsish for property 'Name' throws at mydomain/auraFW/javascript/mhontaYdOya4Y_lBu7v9yg/aura_prod.js:2:27687  HTML Code:   &lt;template&gt;      &lt;template if:true={wiredContact}&gt;          {wiredContact.Name}          &lt;lightning-input value={wiredContact.Name} onchange={updateName}&gt;&lt;/lightning-input&gt;     &lt;/template&gt;  &lt;/template&gt;   JS:   import { LightningElement ,wire,track,api } from 'lwc'; import myContact from "@salesforce/apex/ContactController.fetchContact";  export default class Myrefreshapextest extends LightningElement {    @track wiredContact;   @wire (myContact)         fetchedContact({error, data}){             if(data){                 console.log(JSON.stringify(data));                 this.wiredContact = data;             }else if (error){                 console.log(error);             }     }      updateName (event){         console.log(JSON.stringify(event.detail.value));         console.log(JSON.stringify(this.wiredContact));         this.wiredContact.Name = event.detail.value;     }  }   Apex:  public class ContactController {      @AuraEnabled(cacheable=true)     public static Contact fetchContact(){         return [SELECT Id,Name FROM COntact LIMIT  1];     } }   On top of my head, am not doing anything wrong, anyone has idea what's wrong with my code?  When I print console.log(JSON.stringify(this.wiredContact)); I get old values so am pretty sure it exists.  I tried with @track and @api, but same response. Can anyone shed some light? I have an input type="image". This acts like the cell notes in Microsoft Excel. If someone enters a number into the text box that this input-image is paired with, I setup an event handler for the input-image. Then when the user clicks the image, they get a little popup to add some notes to the data.  My problem is that when a user enters a zero into the text box, I need to disable the input-image's event handler. I have tried the following, but to no avail.  $('#myimage').click(function { return false; }); Area of a quadrilateral proof Is "mens" a valid word? I've been living in Ireland for almost a year now and I start noticing they use the word "mens" a lot. I can see it used in:   Shops, to denote the area where you can find men's clothes In sport, when they talk about "mens team".    My guess is just that they are lazy about the use of quotes, so that mens should actually be men's. However, there may be some rule I'm not aware of. So, is "mens" only limited to Irish English? When I'm allowed to use it? Find the radius of the inscribed circle of a right triangle. Find the radius of the inscribed circle of a right triangle. The triangle's height is $\sqrt{6} + \sqrt{2}$ while the bisector of the right angle is 4.  This seemed like a generic similar triangles problem but it isn't that simple when I tried to solve it. Any help?  EDIT: Here is a picture of the problem Please give me feedback on my answer to this question. Question: For all $ n\geq1:\binom{2n}{0}+\binom{2n}{2}+\binom{2n}{4}+\cdots+\binom{2n}{2k}+\cdots+\binom{2n}{2n} $   is equal to $ \binom{2n}{1}+\binom{2n}{3}+\binom{2n}{5}+\cdots+\binom{2n}{2k+1}+\cdots+\binom{2n}{2n-1} $. Answer: False, since;  Let $a=1,  b=-1$  and $n=1$  ,$k=0$   , using Binomial expression.  $$(a+b)^n=  \sum   \binom{n}{k}  a^{k}b^{n-k}$$  Then $(1+-1)^{2(1)}=\binom{2}{0}1^{0}(-1)^{2-0}+\binom{2}{1}1^{1}(-1)^{2-1}+\binom{2}{2}1^{2}(-1)^{2-2}=\binom{2}{0}-\binom{2}{1}+\binom{2}{2}$  Thus :For all $n\geq1:  \binom{2n}{0}+\binom{2n}{2}+\binom{2n}{4}+\cdots+\binom{2n}{2k}+\cdots+\binom{2n}{2n}   \ne \binom{2n}{1}+\binom{2n}{3}+\binom{2n}{5}+\cdots+\binom{2n}{2k+1}+\cdots+\binom{2n}{2n-1}$  . I have a configuration file where a developer can specify a text color by passing in a string:   &lt;text value="Hello, World" color="Red"/&gt;   Rather than have a gigantic switch statement look for all of the possible colors, it'd be nice to just use the properties in the class System.Drawing.Brushes instead so internally I can say something like:   Brush color = Brushes.Black;   // Default   // later on...  this.color = (Brush)Enum.Parse(typeof(Brush), prasedValue("color"));   Except that the values in Brush/Brushes aren't enums.  So Enum.Parse gives me no joy.  Suggestions? I have a WinForms app that I am trying to make full screen (somewhat like what VS does in full screen mode).  Currently I am setting FormBorderStyle to None and WindowState to Maximized which gives me a little more space, but it doesn't cover over the taskbar if it is visible.  What do I need to do to use that space as well?  For bonus points, is there something I can do to make my MenuStrip autohide to give up that space as well? Find the functions $f:\Bbb R_*^+ \to \Bbb R_*^+$ such that:  $$f(x)f \left(\frac{1}{x}\right)=1$$