Let $r&gt;0$ be a rational number. There is a criterion for determining whether the equation $x^2+y^2=r$ has a solution in rational numbers, and if we have one solution, then there is a way to generate all solutions. However, how does one find the first solution?
Which rational numbers can be written as the sum of two rational squares?  That is, for which rational numbers $a$, are there rational numbers $x$ and $y$ such that $a = x^2 + y^2$.  It is a famous theorem that if an integer can be written as the sum of two rational squares then it can be written as the sum of two integral squares, and then the solution is the famous one by Fermat, but I didn't find anything about the general case.
Wacom Tablet - Illustrator brush tool I currently got the intuos art wacom tablet so still learning how to use it. I'm currently in illustrator trying to draw out a map and the brush tool is doing something weird and I cannot figure out why. You can see in my image that where there are curves the black slightly spills over and is filling in half the curve with black. Brush setting is set up as a new calligraphic brush and the size is set to pressure, 3pt-3pt. Which words in a title should be capitalized? Are there any concrete rules that say which words (parts of speech) in a title should start with a capital letter? What would be a correct capitalization for the title of this question? Definition of electric charge and proper explanation Bmp to jpg/png in C# Show that $n$ lines separate the plane into $\frac{n^2+n+2}{2}$ regions How can I add a program to the context menu of all files? Bring Debian to second Position in the GRUB Menu at startup Baby Rudin Theorem 1.20 (b) Proof I have a question about Rudin's proof of Theorem 1.20 (b) in his book Principles of Mathematical Analysis.  Theorem 1.20 is stated as follows:     (a) If $x\in R, y\in R$, and $x&gt;0$, then there is a positive integer $n$ such that $$nx&gt;y.$$   (b) If $x\in R, y\in R$, and $x&lt;y$, then there exists a $p\in Q$ such that $x&lt;p&lt;y$.     I understand Rudin's proof of (a).  The beginning of Rudin's proof of (b) is given below:     Since $x&lt;y$, we have $y-x&gt;0$, and (a) furnishes a positive integer $n$ such that $$n(y-x)&gt;1.$$  Apply (a) again, to obtain positive integers $m_1$ and $m_2$ such that $m_1&gt;nx$, $m_2&gt;-nx$.  Then $$-m_2&lt;nx&lt;m_1.$$  Hence there is an integer $m$ (with $-m_2\leq m\leq m_1$) such that $$m-1\leq nx&lt;m.$$   I don't understand the justification for this last sentence beginning "Hence...."  How is $m$ found, and why are $m_1$ and $m_2$ needed to find $m$? Inequality: $ab^2+bc^2+ca^2 \le 4$, when $a+b+c=3$. Proof for two sequences producing the maximum value when sorted I am trying to improve my discrete Math skills by doing some proof exercises. But I am struggling to understand how to start proving the following hypothesis:  $$ S = {x}_1{y}_1 + {x}_2{y}_2 + \ldots + {x}_n{y}_n $$ Where ${x}_1,{x}_2\ldots{x}_n$ and ${y}_1,{y}_2\ldots{y}_n $ are ordering of two different sequences of positive real numbers, each containing n elements. S takes its maximum value when ${x}_1,{x}_2\ldots{x}_n$ and ${y}_1,{y}_2\ldots{y}_n $ are sorted in non decreasing order.  I was thinking that I should maybe use a proof by contradiction. Where the negation of the hypothesis is: S takes it minimum value over all ordering of the two sequences when both sequences are sorted into non increasing order.  However I am not sure my negation of the hypothesis is correct or how proceed from there. I do not want the complete proof because I am trying to learn how to do it myself but if some one can tell me how to make a start on this I would be grateful. Imaginary line on the Earth is straight or not? I was watching Google map. Saw a line which is joinging north to south pole. It is going between middle of China and Mexico. But, the line is not straight. Tried to search in net but not able to get anything. Can anyone check and let me know  or Prove that all 2x2 orthogonal matrices can be expressed as Rotation or Reflection The Windows 8 version of Google Chrome seems to have been updated to support the UI formerly known as Metro.  However, this seems to be the only option now, which means that I can no longer use Chrome on my netbook.  How do I get Google chrome to work in classic desktop mode on Windows 8?  Thanks "Dare" with and without "to" To my surprise, there's a missing question about this particularly interesting verb, dare. All I know about it is the fact it can be in two forms, as an auxiliary (without to: "I dare not mention their names") and main (with to: "Did anyone dare to admit it?") verb and the difference has something to do with agreeing with subject. Can somebody explain in a greater detail? Is it possible to make an underfull tabulary spread out? Does javascript use immutable or mutable strings? Do I need a "string builder"? Where to place and how to read configuration resource files in servlet based application? Python Split String Lets Say we have Zaptoit:685158:zaptoit@hotmail.com   How do you split so it only be left 685158:zaptoit@hotmail.com Why can lstlisting not be saved in a command? How do I prove that a graph is a topological manifold I'm having problems with that:  Prove that M is a topological manifold.  $f:U \to \mathbb R^k$, U $\subset \mathbb R^{n}$ open, f continuous  M = $\{\left(x,y\right)\in \mathbb R^{n+k} \mid x \in U, y=f(x) \}$  Can anyone help me please?