Is it possible to proof Euclid theorem which stated that "there are infinitely many primes" using Sieve theory? is a set of general techniques in number theory, designed to count, or more realistically to estimate the size of, sifted sets of integers. The primordial example of a sifted set is the set of prime numbers up to some prescribed limit X. Correspondingly, the primordial example of a sieve is the sieve of Eratosthenesthat there are infinitely many primes, One successful approach is to approximate a specific sifted set of numbers as example the set of prime number.  Really i'm interested in this question :     Question: Is it possible to prove  which stated that "there are infinitely many primes" using Sieve theory ?
Sieve of eratosthenes infinite primes I don't understand why the sieve of eratosthenes doesn't prove an infinite number of primes.  For example the number 2 eliminates 1/2 of an infinite number of numbers.  The next prime, 3, eliminates 1/3 of the remainder, not 1/3 of all the numbers and so on.  5 would eliminate 1/5 of the remaining numbers. Why doesn't that prove an infinite number of primes? Thanks
3 prong to 4 prong dryer cord change My 2 year old whirlpool dryer has a 3 prong cord. My new house has a 4 prong. The dryer doesn't have a spot for 4 wires. The ground wire coming from the dryer is attached to the dryer itself. A new 4 pronged dryer cord has white, red, black, and green wires. Maybe the green wire hooks to the ground screw with the one coming from the dryer? How do I show a function on 2-adic units is continuous (using the 2-adic metric)?   I'd be happy to learn the general rule or definition. But in particular I need to show that $f(x)=\dfrac{3x+1}{2^{v_2(3x+1)}}$ is continuous at all odd numbers.  Since the function is isometric in $\lvert\cdot\rvert_2$, i.e. since $\lvert f(x)\rvert_2=\lvert x\rvert_2$ every orbit of the function is duplicated, multiplied by any power of $2$.  This can therefore be formulated in various ways; as a function though the odd numbers, through the dyadic rationals, but the following seems good to work with.  If we let $n$ be a positive integer and $k$ be the power of $2$ we can define:  $f(2^k(2n+1))=(3(2n+1)+1)\cdot2^k\cdot\lvert3(2n+1)+1\rvert_2$  So $f(2^k(2n+1)=(3n+2)\times2^k\times\lvert3n+2\rvert_2$  Here is what I have so far:  I think to prove continuity in the 2-adic metric I need $\lvert x_n-x\rvert_2=0\implies\lvert f(x_n)-f(x)\rvert_2=0$  I think $k$ and $n$ are independent so I think I can examine them independently.  Taking $k$ to infinity brings both $x$ and $f(x)$ to zero so that seems to satisfies the continuity requirement.  Moving on to $n$; this seems to be an exercise in proving convergence within odd integers which are in a sense a subset of the 2-adic units.  I know all Cauchy sequences in these converge to 2-adic units but not much more than that.  However I do have a little insight into this particular function.  For example if we examine the inputs $x$ which map to any given output of $f(x)$, it can fairly easily be shown that these $x$'s take the form of a set $\left\{4^mp+\dfrac{4^m-1}{3}:m,n\in\mathbb{N}\right\}$ so, at least for any given output $f(x)$ the inputs always converge to $x=\frac{-1}{3}$ as the intervals between them become large powers of $2$.  In fact the orbit of the function $a(2^kx)=2^k(4x+1)$ on variation of $x$ and holding $k$ fixed is in a sense orthogonal to $f(x)$; which is a restatement of the above except not in closed form. Graphics horribly corrupted on Ubuntu 16.04 in VirtualBox, showing random strips of pixelated colors When does a Square Matrix have an LU Decomposition? Show that $8 \mid (a^2-b^2)$ for $a$ and $b$ both odd "Speed" of Gravity and Speed of Light Some threads here touching speed of gravity made me think about that. This lead to some questions.    The speed of gravity was not measured until today (at least there are no undebated papers to that effect). It seems common knowledge/belief among physicists that the speed of gravity is the same as the speed of light.    And this is my question: Is that speed of light = speed of gravity a result of GR or is that fed into the theory? Or is there some evidence from other sources than GR for this? Does the same speed imply some deep-lying relation between gravity and electromagnetism? Why hash the message before signing it with RSA? Python scope: "UnboundLocalError: local variable 'c' referenced before assignment" There was a film from the 80s in which young people can teleport anywhere they can think of.   These guys are recruited to do studies with, maybe by the government or a secret company which wants the take advantage of their abilities.   I saw it on cable TV around 1990-1997.   Any clues? No matter how nice the composition, flow, and lines of my design are, if I am not copying directly from nature, I have a very difficult time picking a decent color scheme.  It either looks too drab, or too garish.  Is there a good resource that will help me to choose a color scheme that has 2 or 3 colors, and the colors do not clash?  I'm not looking for specific colors or color schemes.  What I need is a resource that will help me choose a color (or colors) that go nicely with, or provide a good accent to, whatever main color I am working with on the design. How to sort strings in JavaScript I have a list of objects I wish to sort based on a field attr of type string. I tried using -  list.sort(function (a, b) {     return a.attr - b.attr })   but found that - doesn't appear to work with strings in JavaScript. How can I sort a list of objects based on an attribute with type string? I was writing a program in C++ to find all solutions of ab = c, where a, b and c together use all the digits 0-9 exactly once. The program looped over values of a and b, and it ran a digit-counting routine each time on a, b and ab to check if the digits condition was satisfied.  However, spurious solutions can be generated when ab overflows the integer limit. I ended up checking for this using code like:  unsigned long b, c, c_test; ... c_test=c*b;         // Possible overflow if (c_test/b != c) {/* There has been an overflow*/} else c=c_test;      // No overflow   Is there a better way of testing for overflow? I know that some chips have an internal flag that is set when overflow occurs, but I've never seen it accessed through C or C++.    Beware that signed int overflow is undefined behaviour in C and C++, and thus you have to detect it without actually causing it. For signed int overflow before addition, see . How do I add a directory to the $PATH in Ubuntu and make the changes permanent? Find all polynomials $P:\mathbb{C}\rightarrow\mathbb{C}$ such that $$P(x^2)=P(x)^2 .$$   Here is what I tried:  First, it is easy to see the constant solutions, namely $P\equiv 0,P\equiv 1$.  Let $r$ be a root of $P$ (i.e. $P(r)=0$). It follows that all terms in the infinite sequence $r,r^2,r^4,\dots,r^{2n},\dots$ are roots of $P$. To avoid a polynomial with infinite roots, we have to have the sequence be periodic. So it turns out that either $r=0$, or the roots of $P$ are roots of unity of degree $2n$ for some $n\in\mathbb{N}$.  In the first case, we get $P(x)=xQ(x)$. Plugging in, we get $$x^2Q(x^2)=x^2Q(x)^2\implies Q(x^2)=Q(x)^2 ,$$ i.e. that $Q$ satisfies the same condition as $P$.  In the second case, we have $P(x)=(x^{2n}-1)Q(x)$. Plugging in, we get $(x^{2n}-1)Q(x^2)=(x^{2n}-2x^n+1)Q(x)^2$. I was not sure how to proceed from here. How can I make a script that opens terminal windows and executes commands in them? I have three scripts I need to run when I start up my Ubuntu machine, they start services I use in my development environment.  To do that, I manually open three terminals and type in the commands.  Is there any way to create a script that will open three terminals and execute one command in each of these? (Each command should be in a separate terminal window so I can see their output). Imgur is over capacity - some user avatars fail to load I see a lot of 'white' avatars in the question lists, e.g.  and . Some user avatars aren't loaded, e.g.  and  – they just timeout for me. Sometimes (but not always) I do get a nice "Imgur is over capacity!" image, which I'm (rather surprisingly) able to upload to imgur - but you might not see it. How to stop HTTPS requests for non-ssl-enabled virtual hosts from going to the first ssl-enabled virtualhost (Apache-SNI) I hope that title is clear.  How do I prevent HTTPS requests for non-ssl-enabled virtual hosts from going to the first ssl-enabled virtualhost (setup is Apache-SNI).  For example, using my abbreviated config below, requests for https://example.com (a non-ssl vhost) are being served by Apache at the ssl-enabled vhost https://example.org. I'd like to disable that behavior and possibly reply with the appropriate HTTP response code (unsure of what that is).  It may not even be possible, but I thought I'd ask.  # I actually have a SNI setup, but it's not demonstrated here. # I don't think it's relevant in this situation.  NameVirtualHost *:80 NameVirtualHost *:443  &lt;VirtualHost *:80&gt;     ServerName example.org &lt;/VirtualHost&gt;  &lt;VirtualHost *:443&gt;     ServerName example.org &lt;/VirtualHost&gt;  &lt;VirtualHost *:80&gt;     ServerName example.com &lt;/VirtualHost&gt;   EDIT: Maybe a mod_rewrite rule in the first ssl-vhost? Single word for floating dust visible in sunlight Is there a single word used in English for the visibility of dust particles floating in a stream of sunlight?     shutterstock.com Java reading a file into an ArrayList? Proving $T$ and $U$ are equal if I know $(X,T)$ and $(X,U)$ are hausdorff and compact This question is from Wayne Patty Exercise 4.2  Let $T$ and $U$ are Topologies on set X. Suppose $ (X,T) $ and $(X,U)$ are compact Haurdsorff Spaces. Prove that $T=U$ or $T$ is not a subset of $U$ or $U$ is not a subset of T.  I thought of proving it by this way: Assuming that if $T$ is a subset of $U$ or $U$ is a subset of $ T$ then $T=U$ must be true. But I was unable to think on how to use compactness or hausdorff property and was unable to proceed. Well, also this question was asked before on this website. But I am unable to understand how the solution will work despite trying to understand it for a fair amount of time as it was just outlined. So, I am looking for a detailed solution.