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divisibility question in abstract algebra over a field $d|n \Rightarrow x^{p^d} - x$ divides $x^{p^n} - x$ over $\mathbb{F}_{p}$ where $\mathbb{F}$ is a field.  Attempt:   $d|n \Rightarrow x^{p^d} - x$ divides $x^{p^n} - x \Rightarrow x^{p^d - 1} - 1$ divides $x^{p^n - 1} - 1 \Rightarrow p^d - 1$ divides $p^n - 1$ since $p^n - 1 = (p^d)^{\frac{n}{d}}-1$.  How does this look? Find Items in one column that are not in another column I have two columns in Excel, and I want to find (preferably highlight) the items that are in column B but not in column A.   What's the quickest way to do this? Usage of into vs in vs inside When should 'into', 'in' and 'inside' be used?  What are the differences between the three?   For example, what are the differences between the following statements?  Are any of them incorrect?        The frog jumped into the well.   The frog jumped in the well.   The frog jumped inside the well. Evaluate $\lim_{x \to 0} (x\lfloor\frac{1}{x}\rfloor)$ Cancellation of Direct Products Why would I want to use Interfaces? I understand that they force you to implement methods and such but what I cant understand is why you would want to use them. Can anybody give me a good example or explanation on why I would want to implement this. Perfect infinitive Intuition behind the Definition of Conditional Probability (for 2 Events) What is some intuitive insight regarding the conditional probability definition: $P(A\mid B) = \large \frac{P(A \cap B)}{P(B)}$ ? I am looking for an intuitive motivation. My textbook merely gives a definition, but no true development of that definition. Hopefully that's not too much to ask. $x^4 + y^4 = z^2$ $x, y, z \in \mathbb{N}$,    $\gcd(x, y) = 1$  prove that $x^4 + y^4 = z^2$ has no solutions.  It is true even without $\gcd(x, y) = 1$, but it is easy to see that  $\gcd(x, y)$ must be $1$ The first day of the current month in php using date_modify as DateTime object Significance of assumption in competition inequality questions Refer to the problem below (IMO 2009 Shortlist)     Let $a, b, c$ be positive real numbers such that $$\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = a + b+ c$$   Prove that    $$\frac{1}{(2a+b+c)^2} + \frac{1}{(2a+b+c)^2} + \frac{1}{(2a+b+c)^2} \leqslant \frac{3}{16} $$    In the solution given by the  (Pg 16), it states that      Without loss of generality, we choose $$a +b+c = 1$$.    Thus, the problem becomes    $$ \frac{1}{(1 + a)^2} + \frac{1}{(1 + b)^2} +\frac{1}{(1 + b)^2} \leqslant \frac{3}{16}(\frac{1}{a} + \frac{1}{b} + \frac{1}{c})$$    Applying Jensen’s inequality to the function $$f(x) = \frac{x}{ (1+ x)^2}$$   ,which is concave for $0 ≤ x ≤ 2$   and increasing for $0 ≤ x ≤ 1$, we obtain   $$α \frac{a}{(1 + a)^2} + β \frac{b}{(1 + b)^2} + γ \frac{c}{(1 + c)^2} \leqslant (α + β + γ) \frac{A}{(1 + A)^2}$$   , where $A =\frac{αa + βb + γc}{α + β + γ}.$   Choosing $α = \frac{1}{a} , β =\frac{1}{b},$ and  $γ = \frac{1}{c}$   , we can apply the harmonic-arithmetic-mean inequality   $$A =\frac{3}{\frac{1}{a} + \frac{1}{b} + \frac{1}{c}} ≤ \frac{a + b + c }{3} = \frac{1}{3} &lt; 1$$   Finally we prove:   $$ \frac{1}{(1 + a)^2} + \frac{1}{(1 + b)^2} +\frac{1}{(1 + b)^2} \leqslant ( \frac{1}{a} + \frac{1}{b} + \frac{1}{c}) \frac{A}{(1 + A)^2} \leqslant ( \frac{1}{a} + \frac{1}{b} + \frac{1}{c})\frac{A}{(1 + \frac{1}{3})^2} = \frac{3}{16}(\frac{1}{a} + \frac{1}{b} + \frac{1}{c}) $$   However,  (Pg 4) differs by having $a +b+c=3$ and setting $f(x) = \frac{1}{16x} - \frac{1}{(x+3)^2}$ and allowing Jensen to resolve the rest.   The questions are as follows;   How do you choose before hand which $a + b+ c =$ to choose?  Why particularly does the assumption differ between the 2 solutions? How does one know before hand the choosing of $α = \frac{1}{a} , β =\frac{1}{b},$ and  $γ = \frac{1}{c}$?  Where and how did $A =\frac{αa + βb + γc}{α + β + γ}$ come from?   Any help would be much appreciated. How can I classify data with the nearest-neighbor algorithm using Python? I am trying to show that if $n$ is an odd natural number, then $8$ divides $n^{2}-1.$ I was able to prove that because I know that if $n$ is an odd natural number, then $n^{2}$ can be written as $8k+1$ for some $k\in \mathbb{N}.$ I would like to show this question by using the Euclidian division. Then I wrote $n^{2}-1=8k+r$, where $0\leq r  &lt; 8.$ When $r=2$ we get $n^{2}-1=8k+2.$ Since $n$ is odd, then $n^{2}-1$ is even and I got stuck. Is there way to fix that? Converting Raw HTTP Request into HTTPWebRequest Object I made something like this:    I did this with the Ellipse Tool and the Line Tool.  Now the circle is accurate, but the lines inside are very inaccurate. Is there another tool that does the same thing but with a lot more accuracy? Bound states of the $V(x)=\pm \delta'^{(n)}(x)$ potential? The $\delta(x)$ Dirac delta is not the only "point-supported" potential that we can integrate; in principle all their derivatives $\delta', \delta'', ...$ exist also, do they?  If yes, can we look for bound states in any of these $\delta'^{(n)}(x)$ potentials? Are there explicit formulae for them (and for the scattering states)?  To be more precise, I am asking for explicit solutions of the 1D Schroedinger equation with point potential,  $$- {\hbar^2 \over 2m} \Psi_n''(x) + a \  \delta'^{(n)}(x) \Psi(x) \ = E_n \Psi_n(x) $$  I should add that I have read at least of three set of boundary conditions that are said to be particular solutions:   $\Psi'(0^+)-\Psi'(0^-)= A \Psi(0)$ with $\Psi(0)$ continuous, is the zero-th derivative case, the "delta potential". $\Psi(0^+)-\Psi(0^-)= B \Psi'(0)$ with $\Psi'(0)$ continuous, was called "the delta prime potential" by Holden. $\lambda \Psi'(0^+)=\Psi'(0^-)$ and $\Psi(0^+)=\lambda\Psi(0^-)$ simultaneusly, was called "the delta prime potential" by Kurasov   The zero-th derivative case, $V(x)=a \delta(x)$ is a typical textbook example, pretty nice because it only has a bound state, for negative $a$, and it acts as a kind of barrier for positive $a$. So it is interesting to ask for other values of $n$ and of course for the general case and if it offers more bound states or other properties. Is it even possible to consider $n$ beyond the first derivative?  Related questions  (If you recall a related question, feel free to suggest it in the comments, or even to edit directly if you have privileges for it)  For the delta prime, including velocity-dependent potentials, the question has been asked in   In the halfline $r&gt;0$, the delta is called "Fermi Pseudopotential". As of today I can not see questions about it, but  seems to be the same potential.  A general way of taking with boundaring conditions is via the theory of self-adjoint extensions of hermitian operators. This case is not very different of the "particle in 1D box", question  A general take was the question  A related but very exotic question is  because obviosly the point-supported interacctions have a peculiar scaling  Comments  Of course upgrading distributions to look as operators in $L^2$ is delicate, and it goes worse for derivatives of distributions when you consider its evaluation $&lt;\phi | \rho(x) \psi&gt;$. Consider the case $\rho(x) = \delta'(x) = \delta(x) {d\over dx}$. Should the derivative apply to $\psi$ only, or to the product $\phi^*\psi$? How to get rid of banding in illustrator while using gradient? How to get Rid of Banding in illustrator while using Gradient How to use Whatsapp mobile application on Ubuntu Touch? In my Android I have "Whatsapp" application running. But mobile freak I am not, and specially the mobile keyboards.  So how can I run my same application, same account from my Ubuntu desktop PC?   Follow up:  Just to do keyboard routing, i have used . Which allows me to takeover the Android input fields access via web link to type in my PC keyboard, and then it re-route the keys to Mobile.  Features:   Allows me to copy and paste large texts to Android input fields without using the keyboard of Android but simply using my Desktop PC, via web link http://android.ip.address.goes.here:port/ Allows me to have key submit which is similar to key press of mobile set itself   At-least works without having mobile in rooted. Hope it helps you!! Question  I want to write a book in Arabic (I have an Arabic keyboard, so I don't need a converter). By Arabic book I mean literally Arabic:    title, table of contents  etc    all in Arabic.   The equations, however, I want to write them in English/Latin.  Related    This link helped a lot but it was for Arabic texts in an English main document.    I already checked these two (and plenty other sites). Even/Odd Binomial Coefficients I was wondering if there's a nice general solution for the following problem:  How many numbers in the $n^\text{th}$ row of Pascal's triangle are even? How many are odd?