Solving $2x^4+2x^2y^2+y^4 = z^2$
Prove that the equation $x^4+(2k^2)x^2+2k^4$ is not a perfect square
Sometimes when attempting to capture a Pokémon, the mon will be wiggling in the ball and the app will disconnect from servers (for mobile connection reasons or whatever), and I'll be left staring at a ball containing Schrodinger's Pokémon. I'll sigh at the loss (or in the case of a 500CP Pinsir, scream in agony), and reload the game.  I thought maybe encounters in general couldn't handle disconnecting well, but sometimes immediately after a Pokémon broke free of a ball, the little icon in the upper left would appear, the game would stall for a bit, but it would recover.   Is there anything I can do to avoid this? Should I be waiting it out, instead of reloading?  (and before anyone says it, yes I have filed a bug report) How to use host names rather than ip addresses on home network? I've a small home network up and running and managed to set up ssh. Can anyone tell me what I need to do to refer to one computer on the network from another by host name rather than IP address (which is what I'm doing now)? This is a repost with additional info of an older question that was never answered. I have stumbled upon this old question while also trying to find the name of the book series myself, as I dearly wish to read it again but for the life of me, I can't remember the name of it.      This is a long shot but I'm really in need of help for this because      it was an amazing book. It was either a kid's or YA book. A guess      from what year it was published would be 90s-mid 2000s, as I read it      around 2008-2011.         The book cover was in first person view, with the main character   holding a weapon, and was a 3D rendered image inside a dungeon or   cave (that was very reminiscent of early 90s CGI animation), though   there may be variants.    It would have been set around when computer   games were mainstream (maybe around the same time as when it might've   been published) It's about a teenage boy who is bored at home, so he   looks through an old family computer to play a game and finds one   that he has never seen before. When he plays it he gets transported   into the game which is a fantasy-medieval setting.    All that I can   remember about the game and story is that the main character is on a   quest, he rides horses a lot, (to the point where he can ride horses   perfectly in real life), at one point he eats food at an ingame   tavern/restaurant? and he tosses the ingame coin currency at the   server because he's in a hurry, (don't know why I remember that bit   specifically haha), and he meets a gnome/dwarf character who is   helping him on his journey, who has a home in an ingame mountain, and   that time goes faster in the game. (For example, if he spent 5   minutes in the real world, 3 in-game days would have passed)   Sometimes as he's playing something stops him from being able the   play the game for a few weeks, maybe a concerned parent because he's   playing too much or there's a family vacation. I remember during the   time when he's not playing the game he's on a farm and knew how to   ride the horses there.    There was a character in the real world who   was the main character's sibling, relative or friend. They were like   a rival and was also very whiny (to some extent). They were on the   farm as well with him and questioned how the main character knew how   to ride a horse when he had never done it in real life.   This is all I can remember but if you need more specific information I can try and answer. It is definitely not Space Demons or Demons Don't Dream. If anyone knows anything remotely similar please mention it, I would really appreciate it. Thank you!!            Additional info:    It is not an online game or virtual reality game. It was a normal PC game, and all other characters are (I think), just NPCs.  He woke up for the first time in a field/farm and was super confused about what happened and where he was, as he had never played the game before. He is asked a super serious question before starting the game up for the first time, like "are you sure you wish to play ___?", and then he is transported into the game.  He frequently exited the game, to continue his IRL life. And sometimes, he would lose access to the family desktop PC for a couple of days at a time, and when he went back everyone in the game (NPCs), thought he had disappeared because he was gone for ages. I think once, his family when on holidays and he couldn't take the PC, for example.  It was a book series, not a single book. In later books he would play the game again months later in the real world, and years and years would have passed in the game world.  The plot involved 2 main warring sides/factions/countries of humans, but there were fantasy/magical races and characters involved. I vaguely remember something about each side having official colours, one red and one blue. I think the main character was on the blue side.   The original question was never correctly answered, so I'm posting again to see if anyone can remember.   Books that were ruled out in original post: Eden's Gate &amp; Heir Apparent. The files are "ready to download" or "ready to be downloaded"? Both ways seem to be used quite interchangeably even though the first one doesn't seem grammatical to me (English is, however, my second language so I could be miles off here). What do you think? The default in lyx is to have   I. First part (roman number)   Section 1   1.1 Subsection 1  and I'd like to have   First part (arabic number)   1.1 Section 1  1.1.1 Subsection 1  etc.   Any suggestion? Thanks a lot in advance! I heard a lot about in-body image stabilization recently and I wonder how this technology works. I don't mean the mechanical part (I understand how the sensor moves along different axes to compensate), but the technique in principle:  When I take my 5D III and a lens like the 70-200 2.8 II IS USM with activated stabilization, the IS will constantly move one glas element within the lens in such a way that it directs the cone of light exactly onto the sensor, so that you can always use the full sensor area.  When the cone of light goes through an unstabilized lens and you need to move the sensor - how can it capture the same amount of light at all? The light enters the lens down a fixed axis right through the center of the lens. When the sensor currently moves down, for example, to compensate for a shake, I can't see how it then is still able to capture the full resolution? When a part of the sensor is below default/calm position, then there must be missing the same part on the opposite site?  There's only one way I could image IBIS to work without resolution reduction: if the sensor area was significantly bigger than the area exposed to light, so that there's still sensor available when it currently moves out of its initial position. But wouldn't this (partly) be the same as digital stabilization because the final image gets calculated from the full sensor area minus the not exposed area?  I really would like to understand this and it would be great if someone could "shed some light on this". Minimize the sum of distances between two point and a circle Prove the following integral inequality: $\int_{0}^{1}f(g(x))dx\le\int_{0}^{1}f(x)dx+\int_{0}^{1}g(x)dx$ Suppose $f(x)$ and $g(x)$ are continuous function from $[0,1]\rightarrow [0,1]$, and $f$ is monotone increasing, then how to prove the following inequality: $$\int_{0}^{1}f(g(x))dx\le\int_{0}^{1}f(x)dx+\int_{0}^{1}g(x)dx$$ If $\alpha_{1},\ldots,\alpha_{m}$ are vectors different from zero vector, then there is a linear functional $f$ such that $f(\alpha_{i})\neq 0$ I have to show that the ring $\mathbb{Z}[X]/(X^2-1)$ is not isomorphic with $\mathbb{Z}\times \mathbb{Z}$.    I know that $(\mathbb{Z}\times\mathbb{Z})^*=\{(\pm1,\pm1)\}$, so I thought I should be looking for elements which have inverses in $\mathbb{Z}[X]/(X^2-1)$ and hopefully find more or less than 4. But I didnt succeed, so I need hints. Thanks. I just made a minor , and  the grammar correction was unnecessary.  The original text was     Another factor--sometimes life happens and you can't fly. In the old days you could simply sell your ticket to someone else, now you either have to eat a hefty change fee or lose it outright. That's money in their pockets that they didn't used to get.   It was corrected to:     Another factor--sometimes life happens and you can't fly. In the old days you could simply sell your ticket to someone else, now you either have to eat a hefty change fee or lose it outright. That's money in their pockets that they didn't use to get.    Was this grammatical correction an improvement? Was the original text grammatically correct? Is the edited text grammatically correct? What is the correct way to use infinitive after the verb "help": with or without "to"? What is the correct way to use infinitive after the verb "help": with or without "to"?  For example:     Please, help me to understand this.   or:     Please, help me understand this. A pitcher contains $X + 1$ blue balls and $Y + 1$ red balls. It is known that $X, Y$ are independent random variables, and it is given that $X \sim \mathrm{Poisson}(n), Y \sim\mathrm{Poisson}(2n)$. Calculate the limit: $\displaystyle\lim_{n\to \infty}\mathbb P (X + n = Y) $   I tried to condition on the value of $Y$ and to use the Law of total probability but it did not lead to a solution. ( I tried to sum: $ \sum_{i=0}^\infty\mathbb P(X+n=Y|Y=i)\mathbb P(Y=i) $ and get an expression that I can calculate its limit as n goes to infinity) Find all polynomials $P:\mathbb{C}\rightarrow\mathbb{C}$ such that $$P(x^2)=P(x)^2 .$$   Here is what I tried:  First, it is easy to see the constant solutions, namely $P\equiv 0,P\equiv 1$.  Let $r$ be a root of $P$ (i.e. $P(r)=0$). It follows that all terms in the infinite sequence $r,r^2,r^4,\dots,r^{2n},\dots$ are roots of $P$. To avoid a polynomial with infinite roots, we have to have the sequence be periodic. So it turns out that either $r=0$, or the roots of $P$ are roots of unity of degree $2n$ for some $n\in\mathbb{N}$.  In the first case, we get $P(x)=xQ(x)$. Plugging in, we get $$x^2Q(x^2)=x^2Q(x)^2\implies Q(x^2)=Q(x)^2 ,$$ i.e. that $Q$ satisfies the same condition as $P$.  In the second case, we have $P(x)=(x^{2n}-1)Q(x)$. Plugging in, we get $(x^{2n}-1)Q(x^2)=(x^{2n}-2x^n+1)Q(x)^2$. I was not sure how to proceed from here. Motion Tracking objects that go out of frame and re-enter later I am motion tracking a video in which the camera turns away from one scene and then shortly returns to it. However, the segments that went off-screen lose their track and do not continue tracking once the camera returns. I am wondering if there is a manual way to tell a segment where and when to continue looking for it's tracking target. Is this possible to do? Which interpolation technique is suitable for a bathymetry of a small lake? I gave a quick answer to part of  which had not been covered by previous answers, trying to clarify the reason you would say time of decoding but not decoding’s time. I said it was ’s usually indicates possession, but of course there were several counterexamples that would have occurred to me after a moment’s consideration, and these where helpfully supplied:   Britain’s climate two days’ time  a day’s work  the sun’s rays   I am still of a mind to say that possession of some sort is what allows the ’s. Even though the sun does not have title to its rays, they do belong to the sun. Now, at the risk of duplicating the original question and/or being pigheaded, I am curious as to why time of decoding but not decoding’s time is correct, if not for the reason I gave. Difference between Login Shell and Non-Login Shell? I understand the basic difference between an interactive shell and a non-interactive shell.  But what exactly differentiates a login shell from a non-login shell?  Can you give examples for uses of a non-login interactive shell? Why Does Harry Have a Scar When Avada Kedavra Leaves No Trace? Why does Harry have a scar when Avada Kedavra leaves no trace?     ‘The Muggle authorities were perplexed. As far as I am aware, they do not know to this day how the Riddles died, for the Avada Kedavra Curse does not usually leave any sign of damage ... the exception sits before me,’ Dumbledore added, with a nod to Harry’s scar.   Half-Blood Prince - page 343 - Bloomsbury - chapter 17, A Sluggish Memory   What caused Harry to retain a scar from Avada Kedavra, when no one else in know history showed any signs of cause of death when Avada Kedavra was used?  What caused Harry's scar?  I'd love an answer based in canon, such as the books, interviews with J.K. Rowling, or Pottermore. Prove that $z^2 \equiv ab\pmod p$ is solvable if and only if both or neither of $x^2 \equiv a \pmod p$, $x^2 \equiv b \pmod p$ are solvable. Suppose the $p$ is an odd prime not dividing $ab$.  Prove that $z^2 \equiv ab \pmod p$ is solvable if and only if both or neither of $x^2\equiv a\pmod{p}$ and $x^2 \equiv b \pmod p$ are solvable.