How can we prove the theorem that any product of primes in the form $4n+1$ is the sum of 2 relatively prime square numbers?   EDIT: My question is not the same as the responses that were linked. I'm asking specifically about the product of primes in the form $4n+1$ being the sum of $2$ relatively prime squares, not just primes in the form $4n+1$ being the sum of squares.
It is a theorem in elementary number theory that if $p$ is a prime and congruent to 1 mod 4, then it is the sum of two squares. Apparently there is a trick involving arithmetic in the gaussian integers that lets you prove this quickly. Can anyone explain it?
I'm preparing a new template for our students' projects based on memoir. All this projects are supervised (directed) by one or more professors so I would like to declare a command like \author but called \director.   This way the student can declare \director{My professor} and the template will place and format the name.   I would like to be like \author and not just a simple \newcommand because it can include affiliation, several names, etc.  I don't know where is \author declared, where should I look? I want to show that $$\log N&lt;\sum_{n=1}^{N}\frac{1}{n}&lt;1+\log N.$$ But I don't know how to show this. How can I reliably and accurately identify the passive voice in writing or speech? I'm not interested in advice about whether or not to use it yet... I just want to know for sure what it is, so that I don't look as stupid as . Prove that $ \int _a^b\:f\left(x\right)\:\:dx\:=\:\int _a^b\:f\left(a+b-x\right)\:\:dx $ This is the prove- $y=a+b-x$. Then $dy/dx=-1$ and thus $$\int_a^bf(a+b-x)dx= \int_b^a-f(y)dy = \int_a^bf(x)dx$$  But I dont understand the last part of this - Why ? $$\int_b^a-f(y)dy = \int_a^bf(x)dx$$  I know that $\int_b^a-f(y)dy = \int_a^bf(y)dy $. But something is not clear to me. How many RSA keys before a collision? I was wondering how many possible private/public keys exist? If a million people – for whatever reason – would try to generate 5 keys each in the same minute (on the same date and time) is there a high chance of collision? I believe GUID would suffer from that problem as many bits are reversed for date/time (and GUID version) and isn't meant to be used in that way.  Would RSA suffer from collisions if many keys were to be generated in the same moment? Is the amount of possible keys known? I know RSA is based on prime numbers and small numbers are to be rejected. I’m sure values above a certain amount of digits/bits are rejected because software may not be able to support those large values?  So: How many RSA keys before a collision? And if you would try to make many at the same time, would that give you a high chance of collision? Having trouble indexing and executing a query in O (log n) time.  The query includes an INNER JOIN, an ORDER BY, and an equality operation. If I understand the laws of databases correctly, a query can be indexed and executed in O (log n) time (or thereabouts) if a non-equality operator is not used on more than one column. In this case, I believe the INNER JOIN does count as an equality operator and the non-equality operator would be the ORDER BY portion of the query. This table has upwards of 10,000,000 rows and needs to handle several reads and writes per second.  Using PostgreSQL. This is what the table looks like. As you can see, the column 'Names' is a list property and it is the column that the INNER JOIN goes against:  Age Names                       Date 34  ['carla', 'john', 'sam']    3/13/2011 26  ['json', 'cindy', 'joel']   3/13/2011 72  ['beth', 'amber', 'susie']  3/13/2011 14  ['john', 'jim', 'debie']    3/13/2011   This is the query that we are trying to do:  SELECT * FROM the_table WHERE Age==26 AND Names=='john' ORDER BY Date   My background is from using App Engine's Big Table, so I've used equality operators here to indicate that 'john' should be one of the names in the Names column. This would be an acceptable query in GAE's big table, it would execute in O (log N) time as all Big Table queries are reqyured to do. I am assuming there is a way to do this in PostgreSQL as well since PostgreSQL accepts list data types as columns.  Is this possible to do in PostgreSQL?  If so, how should the index be set up (we can't figure out how to set up an index that takes into account the three properties)? $I_m - AB$ is invertible if and only if $I_n - BA$ is invertible. The problem is from  Algebra by Artin, Chapter 1, Miscellaneous Problems, Exercise 8. I have been trying for a long time now to solve it but I am unsuccessful.      Let $A,B$ be $m \times n$ and $n \times m$ matrices. Prove that $I_m - AB$ is invertible if and only if $I_n - BA$ is invertible.    Please provide with only a hint. Application of Liouville's Theorem Let $f(z)$ be an entire function such that $$|f(z)|&lt;\frac{1}{|\text{Im}(z)|},\qquad z\in\Bbb C-\Bbb R.$$ The question asked me to prove that $f(z)=0$. At least looking at it, it really seems to have an application of Liouville's theorem lurking around somewhere, but I haven't found it.   My thoughts first led me to think about doing this by contradiction and using Picard's little theorem. So, I've considered a strip containing the real axis (say of width $2$ for simplicity). This will imply the complement maps into the unit disk, which implies that this strip has to map almost everywhere else on the complex plane. Now, on the imaginary axis, I know that $f(z)$ will vanish as $z\to\infty$, and this seems like it might be useful. I'm sort of under the impression $f(z)$ might have an essential singularity at $\infty$, which would mean $f(1/z)$ has an essential singularity at $0$. Either way, it of course can't have a pole at $\infty$ because of $f(z)$ vanishing on the imaginary axis, and if it is a removable singularity, it must be $0$, which still gives the solution. Therefore, I think it must have something to do with $f(z)$ having an essential singularity at $\infty$.   I was hoping to combine this with the above inequality and deduce a contradiction, but I haven't thought of one yet.   Any hints or suggestions? I'm studying for a prelim. exam, so this isn't homework.  Update: So, in the spirit of searching methods using the essential singularity at $z=0$, we have $$\left|f\left(\frac{1}{z}\right)\right|\leq\frac{1}{\left|\text{Im}\left(\frac{1}{z}\right)\right|}=\frac{|z|^2}{|y|}.$$ Thus, for $|y|&gt;1$, we have $$\left|f\left(\frac{1}{z}\right)\right|&lt;|y|\left|f\left(\frac{1}{z}\right)\right|&lt;|z|^2.$$ Thus, if I can show that $|f(1/z)|$ is a polynomial, we'll know it can't have an essential singularity at the origin, concluding my proof. Entire function bounded by a polynomial is a polynomial Suppose that an entire function $f(z)$ satisfies $\left|f(z)\right|\leq k\left|z\right|^n$ for sufficiently large $\left|z\right|$, where $n\in\mathbb{Z^+}$ and $k&gt;0$ is constant. Show that $f$ is a polynomial of degree at most $n$. How to make blocks in spawn unbreakable in survival? So on my multiplayer server on Minecraft Education Edition I can't seem to figure out how to make the blocks in spawn unbreakable. I know adventure mode fixes that but it's a survival world. I don't want them in adventure, so how do I make the blocks in spawn unbreakable? count vs length vs size in a collection From using a number of programming languages and libraries I have noticed various terms used for the total number of elements in a collection.  The most common seem to be length, count, and size.  eg.  array.length vector.size() collection.count   Is there any preferred term to be used? Does it depend on what type of collection it is? ie. mutable/immutable  Is there a preference for it being a property instead of a method? Proving $\mathbb{F}_p/\langle f(x)\rangle$ with $f(x)$ irreducible of degree $n$ is a field with $p^n$ elements I am attempting to prove what the title says, that $\mathbb{F}_p/\langle f(x)\rangle$ with $f(x)$ irreducible of degree $n$ is a field with $p^n$ elements.  I have already proven that for any field $K$ and polynomial $f(x)$ in $K[x]$, $K[x]/\langle f(x)\rangle$ is a field if and only if $f(x)$ is irreducible in $K[x]$. So I know that $\mathbb{F}_p/\langle f(x)\rangle$ is a field for sure. Where I'm lost is figuring out how to know for sure that there are $p^n$ elements. It's fairly clear if $n\in\{ 0 ,1\}$, but I don't think induction is going to help.  I know from a proposition in class that since $F_p = \mathbb{Z}/\langle p\rangle$ and $F_p = \mathbb{Z}/\langle p\rangle$ is an integral domain, $F_p[x] = (\mathbb{Z}/\langle p\rangle)[x] = (\mathbb{Z}[x])/(\langle p\rangle[x])$, but I'm not sure whether or not that's helpful information. Can you have a class in a struct? (clickable)  Mainad has a valid height and width, however it isn't shown like subad1/subad2. Which are in essence exactly the same! (just a different background image).  Firebug shows my div as greyed out for some weird reason. Replacing the contents of mainad with just some text doesn't solve the problem (problem isn't related to inner elements)..   What's going on?! :( Stop PDFs from displaying inside Google Chrome When I click on a PDF link in Chrome, it opens automatically inside the browser window. How can I get it to download instead and open with an external viewer?  Update: I implemented djhowell's solution but Chrome still seems to be handling PDF files differently from regular files. When I click a PDF link it opens in Adobe Reader, but normally files download to a folder first.  I also find it weird that this is controlled by Reader and not Chrome. Are there not any file-type controls in Chrome anywhere? Prove that an infinite ring with finite quotient rings is an integral domain How can we show that if $R$ is an infinite commutative ring and $R/I$ is finite for every nonzero $I \unlhd R$, then $R$ is an integral domain?   I tried proceeding by contradiction: assume $a$,$b$ $\in R \backslash \{0\}$ and $ab=0$; then $R/(a)$ and $R/(b)$ must be finite, say $R/(a)=\{k_i + (a) : 1 \leq i \leq m\}$ and $R/(b)=\{l_j + (b) : 1 \leq j \leq n\}$. Does this mean $R$ must be finite? Or what about using the fact that $R/(a,b)$ finite?  Thanks for any help with this! How to draw density estimate on histogram by pgfplots? I know how to draw a density histogram using pgfplots from this answer: .  But how can I draw an estimate line on the histogram like this?    \documentclass{article} \usepackage{pgfplots} \begin{document} \pgfplotsset{     small,     every axis plot post/.style={         fill=orange!75,         draw=orange!50!black     },     trim axis left }  \begin{tikzpicture} \begin{axis}[ymin=0, title=\texttt{hist=density}] \addplot [hist=density] table [y index=0] {random.txt}; \end{axis} \end{tikzpicture} \end{document} View approximate number of users online Is there any way to view the number of users online at any given time (or even an approximation)?  This could be really useful to provide an indication of whether your question is likely to get answered any time soon.  It could also provide a bit of an indication for why your question is getting very low views - if there are only a few users online, then that could be the reason for little interest in a question, whereas if there are lots of users online and the question still doesn't garner many views, then maybe it needs to be rephrased or something.  I know that this can't be used definitively, but it could be used as a rough guide, and could also just be an item of interest.  Perhaps it could show the number of users that have been seen in the past 5 / 10 minutes? (Shouldn't be too hard, since each user page already shows the last time that they were seen). A uniformly continuous function maps bounded set to bounded sets I am trying to prove the following:  If $A\subset\mathbb R$ is bounded and $\,f:A\to \mathbb R\,$ is uniformly continuous, then $f[A]$ is bounded.  Could you check my proof?  Let $A \subseteq [-K,K]\subseteq \mathbb R$. Let $\varepsilon = 1$. If $f$ is uniformly continuous there is $\delta$ wuth $|x-y| &lt;\delta$ imply that $|f(x)-f(y)|&lt;1$ for all $x,y\in A$. Let $a \in A$. Then because $A$ is bounded there is a finite number of balls $B(a_n,\delta)$ that cover $A$. Let the number be $N$. Then $f(a)-N \le f(x) \le f(a) + N$ for all $x\in A$. When NASA sends astronauts to space, they have to wear uncomfortable suits. Why aren't the humans in Star Wars spacecraft doing the same thing? What technology do they have on board that remove this need?