I am reading a proof of there are infinitely many primes of the form $4m+3$, but have trouble understanding it. The proof goes like this:    Assume there are finitely many primes, and take $p_k$ to be the largest prime of the form $4m+3$. Let $N_k = 2^2 \cdot 3 \cdot 5 \cdots p_k - 1$, where $p_1=2, p_2=3, p_3=5, \dots$ denotes the sequence of all primes. We find that $N_k$ is congruent to $ 3 \pmod {4}$, so it must have a prime factor of the form $4m+3$, and this prime factor is larger than $p_k$ &mdash; contradiction.  My questions are:   Why is $N_k$ congruent to $3 \pmod{4}$? Why must $N_k$ have a prime factor of the form $4m+3$ if it's congruent to $3 \pmod{4}$?   It seems that those should be obvious, but I don't see it. Any help would be appreciated!
This is the proof from the book:     Theorem. There are infinitely many primes of the form $4n+3$.    Lemma. If $a$ and $b$ are integers, both of the form $4n + 1$, then the product $ab$ is also in this form.  Proof of Theorem: Let assume that there are only a finite number of primes of the form $4n + 3$, say  $$p_0, p_1, p_2, \ldots, p_r.$$ Let $$Q = 4p_1p_2p_3\cdots p_r + 3.$$ Then there is at least one prime in the factorization of $Q$ of the form $4n + 3$. Otherwise, all of these primes would be of the form $4n + 1$, and by the Lemma above, this would imply that $Q$ would also be of this form, which is a contradiction. However, none of the prime $p_0, p_1,\ldots, p_n$ divides $Q$. The prime $3$ does not divide $Q$, for if $3|Q$  then $$3|(Q-3) = 4p_1p_2p_3\cdots p_r,$$ which is a contradiction. Likewise, none of the primes $p_j$ can divides $Q$, because $p_j | Q$ implies $p_j | ( Q - 4p_1p_2\cdots p_r ) = 3$, which is absurd. Hence, there are infinitely many primes of the form $4n +3$.   END  From "however, none of the prime ...." to the end, I totally lost!    My questions:   Is the author assuming $Q$ is prime or is not? Why none of the primes $p_0, p_1,\ldots, p_r$ divide $Q$? Based on what argument?     Can anyone share me a better proof?     Thanks.
Closing jFrame after clicking JButton Why is it so hard to do this in Java? If you want to have any kind of module system you need to be able to load JAR files dynamically. I'm told there's a way of doing it by writing your own ClassLoader, but that's a lot of work for something that should (in my mind at least) be as easy as calling a method with a JAR file as its argument.  Any suggestions for simple code that does this? Quantum teleportation - the alternative to destroying atoms Can't you just disassemble, not destroy our atoms and transport them to another teleporter via networking or telecoms?  This is my thinking of how to keep the same person, not an exact copy of him/her, alive when teleporting.  Please answer to tell me if it's possible or not. Can i prevent/restrict standard user from installing/uninstalling software in Ubuntu 12.04 desktop? Totally bounded, complete $\implies$ compact Can I cast a Warlock spell with another (higher) spell slot from a different multiclass? 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Find the joint density function of $X$ and $Z=X+Y$.   $\begin{aligned}f_{X,X+Y}=&amp;\frac{\partial}{\partial x\partial z}\mathbb P (X\leq x,Y\leq z-x)\\=&amp;\int_{-\infty}^x\int_{-\infty}^{z-x}f(u,v)\,\mathrm dv=\frac{\partial}{\partial z}\left[\frac{\partial}{\partial x}\int_0^x\int_0^{z-x}\lambda^2e^{-\lambda u-\lambda v}\,\mathrm dv\,\mathrm d u\right]\\ =&amp;\frac{\partial}{\partial z}\int_0^{z-x}\lambda^2e^{-\lambda x-\lambda v}\,\mathrm dv=\frac{\partial}{\partial z}\int_x^{z}\lambda^2e^{-\lambda x-\lambda(v-x)}\,\mathrm dv\\ =&amp;\lambda^2e^{-\lambda x-\lambda(z-x)}=\lambda^2e^{-\lambda z} \end{aligned}$  But I don't think this is correct, because $\mathbb P(X\leq x,X+Y\leq z)$ isn't equal to $\mathbb P(X\leq x,Y\leq z-x)$. But I can't think of anything else.  Could someone help me a little bit with this exercise?  I'm going to try the following:  $\mathbb P(X\leq x,Z\leq z)=\int_{-\infty}^x\int_{-\infty}^zf_X(u)f_Z(v)\,\mathrm du\,\mathrm dv$,  where I use the convolution formula. If I made no mistake, this gives  $f_{X+Y}(z)=z\lambda^2e^{-\lambda z}$, if $z&gt;0$.  Then calculating the joint density function:  $\begin{aligned}f_{X,X+Y}(x,z)=\frac{\partial}{\partial z}\left[\int_{-\infty}^z\lambda^3ve^{-\lambda x-\lambda v}\,\mathrm dv\right]. \end{aligned}$  Oh wait, this is also incorrect, because $X$ and $X+Y$ are dependent.  I really don't know how to proceed; could someone help?  I've also looked at this , but they mention the Jacobian, and my teacher has chosen to skip the paragraph about the Jacobian, so it wouldn't make sense if I had to use that. How do I create a completely unattended install of Ubuntu? I need a CD or USB stick, that will install Ubuntu on a completely headless machine. Stick CD in and restart, no keyboard or screen should be involved.  There are a few obstacles:   The language selection menu when you first boot the CD. The fact that the CD menu waits. The installer's asking questions during installation. Limit of a decreasing sequence of measurable sets. Let $(X,\mathcal{A})$ be a measurable space, with measure $\mu$. Let $\{E_n\}_{n \in \mathbb{N}} \subseteq \mathcal{A}$ be a sequence of measurable sets, with $E_{n+1} \subseteq E_n, \ \forall n \in \mathbb{N}$, that is a decreasing sequence, and $\mu(E_n)=+ \infty, \ \forall n \in \mathbb{N}$. Let $E=\bigcap_{n \in \mathbb{N}} E_n$ the limit set of the sequence, $E$ is measurable for definition. Is true that $\mu(E)=+\infty$? If not in all cases, is true with $X=\mathbb{R}^{N}$, $\mathcal{A}$ the collection of Lebesgue-measurable sets, and $\mu$ the Lebesgue measure on $\mathbb{R}^N$? Dropping z dimension programmatically I'm inserting an image with a simple:  \includegraphics[height=5cm]{filename.png}   This results in an image that maintains the aspect ratio, and so gives me something like 6cm width. I only really want the left side of the image though, and would like to crop it to 50% width. Is it possible to do this?  If possible, I'd like to do this in a portable way (i.e. something that other authors of this document won't have to install packages for, and something that works with a reasonably old version of pdflatex - our computers at work are in sore need of an upgrade). I'm having some problems with some scripts in bash, about errors and unexpected behaviors. I would like to investigate the causes of the problems so I can apply fixes. Is there a way I can turn some kind of "debug-mode" for bash, to get more information? I am an Indian student in France on a 6 month visa for an Internship. Currently I don't have my passport with me as I have applied for UK visa and am not hoping to receive it back in next 10 days. I was wondering if I can travel to Germany without my passport in hand.   The documents I can carry are a photocopy of my passport's identity page, Residence proof here, proof of Internship in France,Indian national identity card, receipt from UK embassy stating my visa application.   Please note that I do NOT have a photocopy of French visa page. Debugging using Nvidia Nsight 2.2 in Visual Studio 2010 True displacement in EEVEE? I have a Blender 2.8 scene with animated true displacement. It is created using material nodes and works fine in Cycles. However, EEVEE doesn't displace my meshes truly, only normals are changed. Is it possible and how to solve the problem now or in the future? It seems that panel shadow is somehow hardcoded in KDE 4 and devs do not have any plans to change it and/or make it customizable. So, what I'm actually asking is some kind of hack (if there is any) which will remove the shadow of the panel in KDE 4.x completely (but without need to disable all desktop effects in the process).  Possible application: When a user wants a completely transparent panel, no top/bottom borders at all, the shadow is completely out of place and destroys the experience. How to properly play .mkv files? I am having a problem while playing .mkv files. Every time I play a .mkv file, the audio can be heard but video cannot be seen. The only way to see the video if I actually right click on it or put a window in front of it and it displays one part of the video but not all.    The link of the picture above basically describes what I am saying. While I was trying to take the screenshot, the window for the screenshot disappeared and displayed part of the video. I hope someone could help me on this. Thank you!   EDIT: I was running Unity 2D when I have the problem but when I run Unity 3D, it works. So I need to make this work on 2D. Please help. List.Add seems to be duplicating entries. What's wrong? Is there any simple method to calculate $\sqrt x$ without using logarithm Suppose that we don't know logarithm, then how we would able to calculate $\sqrt x$, where $x$ is a real number? More generally, is there any algorithm to calculate $\sqrt [ n ]{ x } $ without using logarithm? More simple techniques would be nice.  Here is a simple technique used to approximate square roots by :  For example: $\sqrt {78}\approx 8\frac { 14 }{ 17 } $ , where $8$ is the nearest integer root of $78$, $14 = 78 - 8^2$, $17 = 2 \times 8 + 1$.  if $n=2^k$ we can use the method above.   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