Suppose there is an equation $a^2 - b^2 = X$ how many pairs of $(a,b)$ exist to find $X$, when $a,b$ and $X$ are positive integers. It would be great if you could direct to proof although not necessary. An example: $a = 13, b = 11 X = 48$. 13^2 - 11^2 = 48  how many more $(a,b)$ pairs exist for 48? How to find out/them ? I am not a Maths student although i see that this is a variant of a 'diophantine equation'?
Inspired by this Hagon Von Eitzen's answer( ) I started investigating how I could express natural numbers as differences of squares. Using the method explained in the solution, I investigated natural numbers up to $60$ manually. I observed the following:  There is no way to express numbers of the form $4k+2$ as a difference of two squares.  2.There is one and only one way to express prime numbers (except $2$) as a difference of two squares.  The smallest number that can be expressed as a difference of two squares in two different ways is $15$ and the smallest number that can be expressed as a difference of two squares in three different ways is $45$.  As, $15= 8^2-7^2=4^2-1^2$. $45=23^2-22^2=9^2-6^2=7^2-2^2$. I could prove point $2$ easily but could not prove point $1$. Also I am interested in knowing how I could find the smallest natural number that could be expressed as a difference of two squares in four different ways without manual calculation.
$\exp(A)$ is orthogonal iff $A$ is skew symmetric Let $A \in M_n(\mathbb R)$ and $\exp(A)=\sum_{i=0}^{\infty}\frac{A^i}{i!}$. How to prove $\exp(A)$ is an orthogonal matrix iff $A^t=-A$?  I can get that $\exp(A)$ is an orthogonal matrix iff $\exp(A^t)=\exp(-A)$, but how to get $A^t=-A$ from it?  What's more, is $\exp(A)$ injective or not? Conditions for vertical circular motion If a bob attached to a string is swinging in vertical circular motion, we say that the bob leaves the circular path when the tension in the string becomes zero, but even when tension becomes zero, there's always a component of the bob's weight towards the center of the circle. Why then does the bob leave the circular trajectory? IE must close for event with jQuery to work How does a circular lens produce rectangular shots? Is there any way to view Chrome browser notifications history? I get Chrome notifications from several websites and they work when I don't even have any tabs open for those sites.  But sometimes I get these Chrome notifications when I am not at my desk or looking at my laptop.  As some of these notifications are useful to me, I would like to be able to access any that I have missed.  Is there a way to view the notification history? I am afraid that listing doesn't support javascript.  \begin{lstlisting}[language=javascript] ... \end{lstlisting}   What am I doing wrong? Sorry for the bad English. I'm trying to use the tikz-cd package to make commutative diagrams. My code  \documentclass{article} \usepackage[brazil]{babel} \usepackage{tikz-cd} \begin{document} \begin{tikzcd} A \ar[r, "x"] &amp; B \end{tikzcd} \end{document}    The problem occurs when putting a name under the arrow, the syntax "x" is incompatible with the brazil option of the babel package. When exchanging brazil for english it works normally. I already tried using  \usepackage[main=brazil, english]{babel}, \usepackage[brazil, english]{babel} e  \usepackage[brazil, main=english]{babel}   But it did not work. I also tried to use  \usepackage[english, brazil]{babel} \usepackage{polyglossia} \setmainlanguage{brazil} \setotherlanguages{english}   And finally  \selectlanguage{english}   However, it only works if I remove the option brazil UTF-8 encoding a servlet form submission with Tomcat Why can a 1 element set be a member of another set but not a subset of it? I have came across this in a textbook:     $\{2\}\nsubseteq\{\{2\}\}$ but $\{2\}\in\{\{2\}\}$   I understand that $\{2\}$ is an element (member) of the other set but considering $\{2\}$ is a set itself, why is it not a subset? Showing a function on $[0,1]$ is the zero function How to handle OutOfMemoryError in Java? I need a formula for conditional formatting to make an Excel 2010 spreadsheet alternate highlighting when a single column changes.  This question, "", is exactly what I want to do, but the suggested solution didn't work for me--it just highlights the first row of each matching value.  I know how to use conditional formatting, but for the life of me I can't figure out or find any pointers on the net so far to make this work.  Ideas?  Here is a link to a picture of how I want my spreadsheet to look when I'm done.  Basically I want every other Disp Number value row to be highlighted.  Seems like it would be a common thing to want to do, and I've seen this asked for in various places, but people struggle with making it work. I was trying to prove this inequality using induction, but couldn't do.  Question: Suppose $a$ and $b$ are real numbers with $0 &lt; b &lt; a$. Prove that if $n$ is a positive integer, then:  $$a^n-b^n \leq na^{n-1}(a-b)$$ How do I prevent new villager trades from spawning? When making a custom adventure map, usually you only want the trades you programmed into a villager to show up. By default, however, the game will try to spawn new trades when the last one is made. The classic way I'm seeing in tutorials to prevent new trades is to make the last trade untradeable; however, with the release of 1.7, a lot of the old item IDs that were unobtainable have been removed from the game altogether, and following the tutorial I have crashes the game when the trade interface is scrolled to the last trade.   So there's two parts to this question:   Is there a better way to prevent new trades with the 1.7 mechanics? Or with the snapshots of 1.8 mechanics? If not, what is a safe data ID to use to prevent the last trade from being a real trade as of 1.7? Increase close vote weight for gold tag badge holders In order to increase the efficiency with which poor quality questions are closed, it could make sense to have weighted close votes for a small subset of qualified users.   A very good way to measure the qualified users would be to leverage the tag badges. However, it needs to be rationally limited to a set of users that are knowledgeable with the subject. A very good way to measure the top users in a tag would be to look at those who have a gold badge in the tag.  If two users who have a gold tag badge pertinent to a question issue a close vote, the question should probably be closed (or probably will be very soon under the current system). I think it would make sense to allow users with a gold tag badge to have a 3x weight for close votes in those tags.  Please keep in mind, this isn't suggesting full moderator one click closure. It only adds weight to the vote. A single user could not do it on their own. These singular users cast close votes all the time, so that aspect would not change. Some will vote to close, some wont, but the ones who do should have a stronger weight to their vote. Even with 3x it will require 2 of said votes to accomplish, or still 2 other users (meaning 3 agreed it should be closed - which is the same amount of high rep users it takes to delete a question and that has a much greater impact than closure).  tldr;  Allow gold tag badge users to have a weighted close vote in their tag Percentage of overlapping regions of two normal distributions I was wondering, given two normal distributions with $\sigma_1,\ \mu_1$ and $\sigma_2, \ \mu_2$   how can I calculate the percentage of overlapping regions of two distributions? I suppose this problem has a specific name, are you aware of any particular name describing this problem?  Are you aware of any implementation of this (e.g., Java code)? update-initramfs missing /lib/modules/4.4.0-13-generic How to login to gamecenter at PewDiePie's Tuber Simulator? How do I login to the Gamecenter on PewDiePie's Tuber Simulator, if you have not done it when they asked? siunitx tabular numbers with separated uncertainty I would like to format the space reserved for a number shown in a table with separated uncertainty. It seems that when I use the table-format option the brackets around the mantissa disappear. This is illustrated in the example below:  \documentclass{article}     \usepackage{siunitx}  \begin{document}     \begin{table}   \begin{tabular}{                   S[separate-uncertainty = true]                   S[table-format=1.1(1)e1,                   separate-uncertainty = true]                   }     {Header} &amp; {Header}\\     1.0(2)e2 &amp; 1.0(2)e2\\   \end{tabular} \end{table}     \end{document}     I have also tried to use the bracket-numbers and multi-part-units = brackets without solving this issue. Is there any workaround to center the number and keep the brackets like in the first column of the example? I'm tutoring high school students. I've always taught them that:     A charged particle moving without acceleration produces an electric as well as a magnetic field.   It produces an electric field because it's a charge particle. But when it is at rest, it doesn't produce a magnetic field. All of a sudden when it starts moving, it starts producing a magnetic field. Why? What happens to it when it starts moving? What makes it produce a magnetic field when it starts moving?