Positive integer solutions to $3x^4-3x^2+1=y^2$
Can $3p^4-3p^2+1$ be square number?
Limit of $\prod\limits_{k=2}^n\frac{k^3-1}{k^3+1}$ Calculate $$\lim_{n \to \infty} \frac{2^3-1}{2^3+1}\times \frac{3^3-1}{3^3+1}\times \cdots \times\frac{n^3-1}{n^3+1}$$ No idea how to even start. How do I generate a smooth random horizontal 2D tunnel? I'd like to create a smoother version of the navigable and quite natural-looking random tunnel found in .  It should ideally be...   infinite, so more can be generated as the player progresses; parametrised, allowing control of its thickness over time; made of smooth curves, not rectangles as in the above game; precomputable, as knowing its bounds in advance allows collision detection for e.g. positioning powerups inside the tunnel.   I'm looking for a generic method I could implement myself. Further parameters and optimizations are welcome.  Asking here was suggested . I think this fits in both places, as it's as much about the algorithm as about gamedev. Show that if $p$ is prime then $\Bbb Z_p$ is a field Check my proof please.  Let $\Bbb Z_p:=\Bbb Z/p\Bbb Z$ be the quotient ring modulo $p$. I want to prove that if $p$ is prime then $\Bbb Z_p$ is a field.  Known facts about the quotient rings of the kind $\Bbb Z_n$ (I dont list all the characteristics of a ring, just some ones):   They are commutative rings with unity. $[0]=[n]$ is the identity of addition or zero. Then $[n]+[a]=[a]$ and $[n]\cdot[a]=[n]$ for all $[a]\in\Bbb Z_n$. $[1]$ is the multiplicative identity or unity. Then $[1]\cdot[a]=[a]$ for all $[a]\in\Bbb Z_n$. The addition is defined as $[a]+[b]=[a+b]$, and the multiplication is defined as $[a]\cdot[b]=[ab]$. $[a]=[b]$ means that exists $z\in\Bbb Z$ such that $a=b+nz$. $|\Bbb Z_p|=p$.   A ring $\Bbb Z_p$ dont have zero divisors, i.e. doesnt exist $[a],[b]\in\Bbb Z_p$ distinct of $[p]$ such that $[a]\cdot[b]=[p]$. Proof by contradiction: suppose that exists such $[a],[b]$ distinct of $[p]$ that are zero divisors. Then exist some $z_j\in\Bbb Z$ such that:  $$(a+pz_1)(b+pz_2)=pz_3\iff ab+pz_4=pz_3\iff ab=pz_5$$  but this is a contradiction with the assumption that $a,b\neq p$ and $p$ prime, so dont exist divisors of zero on any $\Bbb Z_p$.$\Box$  For any $[a]\in\Bbb Z_p$ distinct of $[p]$ exist $[b]$ such that $[a][b]=[1]$. Because $\Bbb Z_p$ dont have zero divisors and is finite for some $[a]\neq[p]$ we have that if $[b]\neq[c]$ then $[a][b]\neq[a][c]$. Proof by contradiction: if $[a][b]=[a][c]$ in the above conditions then  $$[a][c]=[ac]=[ab]=[a][b]\iff ac+pz_1=ab+pz_2\iff a(c-b)=pz_3$$  for some $z_j\in\Bbb Z$. But $pz_3\in[p]$, then $[a][c-b]=[p]$ but because there are no zero divisors in $\Bbb Z_p$ then or $[a]=[p]$ or $[c-b]=[p]$ what contradicts the above conditions.$\Box$  Because $[a][b]\neq[a][c]$ for $[a]\neq[p]$ and $[b]\neq[c]$ then for any $[a]\neq[p]$ we have that  $$[a]\cdot\Bbb Z_p=\Bbb Z_p\implies \exists [b]\in\Bbb Z_p:[a][b]=[1]$$  In other words: every element of $\Bbb Z_p$ but $[p]$ have a multiplicative inverse as stated above.$\Box$  Because $\Bbb Z_p$ is a commutative ring with $[1]\neq[0]$ and have multiplicative inverse for all their elements but $[p]=[0]$ then $\Bbb Z_p$ is a field.$\Box$ I have to design a circuit in which I have to transfer data through lasers (serial transmission of data). Now what I was thinking was to check if the laser light is on then '1' is passed and when the laser is off '0' would be passed. Now I need to check if a 1 or a 0 was passed.   How would I check for this and another problem I'm facing is if it's on it may keep on passing 1's. How would I check for this? I though of using LDR's (Light Dependent Resistors)? Would this be okay or it there a better method? First, my understanding is that the long-term capital gains tax rate is 0% for those whose marginal rate on ordinary income is 10% or 15%, and (ignoring the highest 39.6% bracket) the rate is 15% for everyone else. (See  and .)  My interest is in what happens at the crucial break between the 15% and 25% bracket, which is $73,800 for married/joint returns in 2013 (and presumably will be higher in 2014, though I haven't been able to find out what that is or if it has even been set yet). Let's say a household has $64,000 in wage income, and sells stock for a $10,000 profit.   Does their income for the purposes of establishing their tax bracket go to $74,000? If so, is the entire $10,000 profit subject to the 15% long-term capital gains rate? Is any part of the wage income subject to the higher marginal tax rate?   Since the amount of income subject to the higher marginal rate is, well, marginal, part 3 seems somewhat less important than part 2. The real issue is whether the capital gains rate hits the entire profit, since, in that case, it seems the household should only sell enough stock to generate a $9000 profit, and sell the remainder in a future year. How to calculate the confidence interval of the mean of means? Imagine that you repeat an experiment three times. In each experiment, you collect triplicate measurements. The triplicates tend to be fairly close together, compared to the differences among the three experimental means. Computing the grand mean is pretty easy. But how can one compute a confidence interval for the grand mean?  Sample data:  Experiment 1:   34, 41, 39  Experiment 2:   45, 51, 52  Experiment 3:   29, 31, 35  Assume that the replicate values within an experiment follow a Gaussian distribution, as does the mean values of each experiment. The SD of variation within an experiment is smaller than the SD among experimental means. Assume also that there is no ordering of the three values in each experiment. The left-to-right order of the three values in each row is entirely arbitrary.  The simple approach is to first compute the mean of each experiment: 38.0, 49.3, and 31.7, and then compute the mean, and its 95% confidence interval, of those three values. Using this method, the grand mean is 39.7 with the 95% confidence interval ranging from 17.4 to 61.9.   The problem with that approach is that it totally ignores the variation among triplicates. I wonder if there isn't a good way to account for that variation. How can I define the different properties of an Cycles/Eevee material? Such things as glossy, glass, diffuse, RGB-color, emissive via python scripting? Placeholder for body off in Android app The placeholder for the body field (the word "Body") is strangely placed in the Android app, seems a bit too high:    Could reproduce using:   Samsung Galaxy 4a.  Android version 4.4.2  App version 1.0.54 Moto X (2013), Android 4.4.4 App version 1.0.60 minimal polynomial of powers of a matrix I am wondering, given the minimal polynomial of $A$, what can we say about the minimal polynomial of $A^n$?  I know that if we were looking at the roots of the characteristic polynomial ie eigenvalues then the roots of the characteristic polynomial of $A^n$ would be $nth$ powers of the eigenvalues so again these are roots of the minimal polynomial also. Is there a way to determine the respective canonical status of Tolkien's Middle-earth related works? It is almost always difficult to determine the canonicity (if that's a word) of different works within a fictional universe, but the problems are even more complicated in regards to Tolkien's Middle-earth/Arda related material.  It is safe to say that LotR, the second edition of The Hobbit, and probably the published Silmarillion are canon.  But even the published Silmarillion is, in large part, the work of Christopher Tolkien, although to a lesser degree than it is the work of his more famous father.    When we look at the rest of the corpus, things become even more opaque.  Unfinished Tales contains different versions of the same stories.  The same is true, in even greater degree, of The History of Middle-earth.  These volumes might be considered an almost equal endeavor on the parts of Christopher and J.R.R Tolkien.    Is there a tentative canon of Tolkien's Middle-earth related works, and if not, is it possible to create one?    With such a large body of work, of varying degrees of canonicity, it would be very helpful to have something like the now-outdated levels of canon associated with the Star Wars universe, ranked by a series of letters.  Has anyone put together such a system for Tolkien's Middle-earth related works? In electrostatics, why the electric field inside a conductor is zero? In electromagnetism books, such as Griffiths or the like, when they talk about the properties of conductors in case of electrostatics they say that the electric field inside a conductor is zero.  I have 2 questions:    We know that conductors (metallic) have free electrons which randomly moves in all directions, so how come we can talk about electrostatics which by definition means stationary charges? When the textbooks try to show why the electric field inside a conductor is zero they say let us put our conductor in an electric field. What happens then is that there will be an induced surface charge density which consequently induces an electric field within the conductor such that the total electric field within the conductor will be zero. That is perfectly understood, but my problem is the following: the original claim was that the electric field within a conductor is 0, not the electric field after putting the conductor in an external electric field it became zero. I do not understand the logic! Where should I put  tags in HTML markup? Where do the files go if you mount a drive to a folder that already contains files? I just accidentally mounted a new drive to a folder that already contained files.  I don't care about them and have them somewhere else, but that folder appears empty now.  I'm curious what happened to the files.  Are they simply deleted by Linux? Conky Widgets are Opening in their Own Windows I just upgraded to 16.04 and I'm getting all my customization set up again, but it seems that most Conky widgets are opening in their own window, instead of just appearing on the Desktop; although some are working correctly.  For example:  LSD (working properly):    Seamod (not working, opening in own window):    Since it's not affecting every single widget, I assume it could be an issue with the conkyrc with the widgets that aren't working, but this also never happened in 14.04 so I wasn't sure if maybe there was some compatibility issue in 16.04. My Google-fu has failed me in finding a solution.  If anyone can point me in the right direction, it would be much appreciated! :) Why does `ls -d *[!e]*` display all files instead of omitting all files that contain an e? ls -d *e* prints out all files containing an "e" somewhere in it. But the opposite, ls -d *[!e]* does not seem to work. Why exactly is that?  is it because ls takes * as its own argument and expands all the files?  I understand I can do: ls -d !(*e*) (in Bash with extglob set, or in ksh). Can a magnetic field be induced without an electric field? Because, as far as I know, a time varying electric field induces a magnetic field an vice versa.  But in the case of conductors carrying currennt, it doesn't seem that electric field varies with time, then how is a magnetic field induced? Leave newlines intact in bounty messages This baby-walking device is called a ([prefix]-)[name] by people from [location] Prove that $C\exp(x)$ is the only set of functions for which $f(x) = f'(x)$ I was wondering on the following and I probably know the answer already: NO. Is there another number with similar properties as $e$? So that the derivative of $\exp(x)$ is the same as the function itself. I can guess that it's probably not, because otherwise $e$ wouldn't be that special, but is there any proof of it? Since Ubuntu 18.04 I can't drag a file e.g. image file to an open application such as Inkscape or  email message window. In Ubuntu 17.10 I was able to select the image file by clicking on it, and while holding down left mouse button I would hit the supper key displaying all open windows where for there I could then drop the file in any open app or folders. This is no longer possible/default behaviour since Ubuntu 18.04. Is there a workaround that would allow me to achieve this or even a way to recreate the same behaviour like in 17.10?