I am really having a problem understanding the proof of this question. Let me write the proof and i will explain my question  Let $p$ be the smallest prime factor of $n$. Assume that $p &gt; n^{\frac{1}{3}}$ Then $p &gt; n^{\frac{1}{3}} \Rightarrow p^{-1} &lt; n^{-\frac{1}{3}} \Rightarrow np^{-1} &lt; nn^{-\frac{1}{3}} \Rightarrow \frac{n}{p} &lt; n^{\frac{2}{3}}$.  Let $m = \frac{n}{p}$, and assume that it is greater than $1$. Notice that $m$ cannot have a prime factor less than $n^{\frac{1}{3}}$. However  $n^{\frac{1}{3}} = \sqrt{n^{\frac{2}{3}}} &gt; \sqrt{m}$  Therefore, $m$ does not have a prime factor less than $\sqrt{m}$. Hence, $m$ is a prime.  I really dont understand this proof and Why does $m$ not have a prime factor less than $n^{\frac{1}{3}}$? If someone can explain to me this proof i would really appreciate it ! thanks
Show that if the smallest prime factor $p$ of the positive integer $n$ exceeds $\sqrt[3]{n}$, then $\frac{n}{p}$ must be prime or 1.  I'm stuck trying to prove this. I tried this using contradiction, but I haven't been able to prove it.  My approach:  Let $p$ be the smallest prime factor of the positive integer $n$.  Let $p &gt; \sqrt[3]{n}$  Then we must show that $\frac{n}{p}$ is either a prime number or 1.  We know that $n = pe$ for some $e \in \mathbb{Z}$.  Thus, $\frac{n}{p} = e$  We also know that $p$ > $\sqrt[3]{n}$, so $p^3 &gt; n$, and $p^2 &gt; \frac{n}{p}$  So, assume $\frac{n}{p} = e$ is not prime nor 1, meaning it's a composite number. Let's try to get a contradiction.  If $\frac{n}{p}$ is composite, then $\exists a,b \neq 1: \frac{n}{p} = ab $  Now, here's where I get stuck. What can I do with this info? $\frac{n}{p} = ab $ and $p^2 &gt; \frac{n}{p}$.
How to scan an entire network using Nmap? Is there a way to scan an entire network using ?   What I want to do is scan my network for all the devices that are currently connected to it. How to run scripts every 5 seconds? I have a script that needs to be run every five seconds. I know that cron can do tasks by the minute, but is there a way to run something every second? 14.04 --> 16.04 failed; apt (>= 1.0.1ubuntu2.13)' is not installed Performing deep learning land cover classification using R? I am really new to Deep Learning and, unfortunately, I can't find example codes on land cover classification other than  where the author wrote a script in R for a large dataset.   The main reason that I am asking is because recently I found a few papers on Remote Sensing Image classification using Deep Learning and I was wondering if there were any R examples on that subject.  So, is there a code example, preferably a step-by-step, which I can use?  I found several git repos like  that can do Deep Learning classification with R, BUTT, I am still looking for a simple multi-spectral image segmentation method. An example of that can be found at  and I also found a similar question in  for which the answer is "use something else other than R if you can".  If I find anything that works today I will be posting a working answer. 2 orderby in wp_query with 2 custom fields I'm triyng to display hotels (taxonomy which i have created) by two 'orderby' with wp_query but i'm not success.  Results looks likes:  Hotel 1 :       2 persons          5 stars Hotel 2 :       2 persons          4 stars Hotel 3 :       2 persons          3 stars Hotel 4 :       3 persons          5 stars Hotel 5 :       5 persons          5 stars Hotel 6:        10 persons         2 stars Hotel 7:        10 persons         1 star   The first 'order by' is on the number of persons(the biggest in first) and the second 'order by' is on the rating in stars (so, if several hotels have the same number of persons, the hotel with the biggest stars rating would be in first).  Below my code. I have tried to add an array inside an array with meta_query but it doesn't work.  $args=             array(                                     'post_type' =&gt; 'hotel',                                     'Services' =&gt; $Services,                                     'Town' =&gt; $Town,                                      'meta_key' =&gt; 'Number_persons',                                     'orderby' =&gt; 'meta_value_num',                                     'order' =&gt; 'ASC',                                        'meta_query' =&gt; array(                                                            array(                                                                'meta_key' =&gt; 'Rating',                                                                'orderby' =&gt; 'meta_value',                                                                'order' =&gt; 'DESC',                                                               )                                                        )                  );   $the_query = new WP_Query( $args );   Thanks a lot :) Aromatic rings resized with chemfig When I want to reduce the size of a molecule drawn with  I usually put a [scale=0.xx] before the argument of the command. However when I deal with aromatic rings this method doesn't work well, in fact only the carbon skeleton is resized: the 'aromaticity' is not and it's also out of center.  For example:  \chemfig[][scale=0.75]{**6(-------)}\quad   gives me:     How do I achieve a complete resize? Radical of $\mathfrak{gl}_n$ Is there a formula for $\sin(xy)$ Can we express a trigonometric function for the product of two angles as a function of trigonometric functions of its factors?  For example: Is there a formula for $\sin(xy)$ as a function of $\sin x$ and $\sin y$ or other trigonometric functions of $x$ and $y$. Showing that $\displaystyle\lim_{s \to{1+}}{(s-1)\zeta(s)}=1$ I need prove the following: ($\zeta(s)$ is the Riemann zeta function)  $\displaystyle\lim_{s \to{1+}}{(s-1)\zeta(s)}=1$  I really don't know, I have tried hard, but have obtained nothing for now. C++ copy-construct construct-and-assign question How did planets have an elliptical orbit in the first place? I know that planets speed up when getting closer to the sun, but they speed up because they have an elliptical orbit, and they have an elliptical orbit because they speed up. Why did they have an elliptical orbit in the first place? I was doing the problem $$ A+B=AB\implies AB=BA. $$  $AB=BA$ means they're invertible, but I can't figure out how to show that $A+B=AB$ implies invertibility. Do there exist general conditions under which we can conclude that continuity on a topological space is detected by $\mathbb{R}$? Whenever $X$ is a topological space, let us say that continuity on $X$ is detected by $\mathbb{R}$ iff for all functions $f : X \rightarrow Y$ where $Y$ is another topological space, we have that if every continuous function $c : \mathbb{R} \rightarrow X$ satisfies "$f \circ c$ is continuous", then $f$ is itself continuous.   Clearly, continuity on $\mathbb{R}$ is detected by $\mathbb{R}$ It is shown  that continuity on $\mathbb{R}^2$ is detected by $\mathbb{R}$. Clearly, it is not the case that continuity on $\mathbb{Q}$ is detected by $\mathbb{R}$, since every continuous mapping $\mathbb{R} \rightarrow \mathbb{Q}$ is constant.      Question. Do there exist general conditions underwhich we can conclude that continuity on a topological space is detected by $\mathbb{R}$? How to make a persistent live Ubuntu USB with more than 4GB I want to make a persistent live Ubuntu USB drive with more than 4GB of storage.   I know there is an option of installing directly on the drive but I don't want to do this as it will restrict the drive from working on different computers.   One option is to make an additional partition with extra space but if I do this how can I get newly installed software to be stored on this partition? Cannot install Ubuntu in VirtualBox due to "this kernel requires an x86-64 CPU, but only detects an i686 CPU, unable to boot" error I was trying to install Ubuntu 12.04 in VirtualBox 4.2.12r84980. I see    this kernel requires an x86-64 CPU, but only detects an i686 CPU, unable to boot   But I am using a 64 bit Windows 8, and trying same .iso for trying Ubuntu. Then what is the problem? Background info: I am planning on making a filehost with which one can encrypt and upload files. To protect the data against any form of hacking, I'd like not to know the encryption key ($K$) used for a file, so the user will have to asymmetrically encrypt $K$ and send it as well. The reason for this double encryption is that you can still share (individual!) files with other accounts by encrypting $K$ with the public keys of those users, too.  My problem is: How can a user can get a public and private key without the server saving/knowing the private key and without the user having to remember a passphrase of 250 hex characters?  Question: Is there a way to derive a key pair from a password, like PBKDF-2 does for symmetric encryption? Suggestions on rearranging the cryptographic steps in an easier way are welcome, too.  Alternatively: It is also possible to encrypt the private key with a PBKDF-applied password and store it on the server, but I'd like to keep things 'simple'. ;)  Note: At this point I don't care if it requires RSA, ECC or an obscure cryptography scheme. I have a registered StackOverflow account which I normally log into using OpenID.  Recently I logged out in order to post a question anonymously (supplying a fake name and of my secondary email addresses).  The problem now is that I'm logged in to StackOverflow as this "dummy" user, unable to see a way to log out.  The top menu normally has a "logout" link, but when you aren't fully registered this link is replaced with "register".  There is no "logout" link for me. :(  So..   When logged out there is a "login" link. When logged in there is a "logout" link. When "logged in" with an unregistered account, there is only a "register" link (no "logout"). How to close entire application when application has more than 2 forms without showing messagebox more than once How to remove Ubuntu from dual boot system with Windows? I wish to remove Ubuntu and I'm not quite sure how. I know it has something to do with partitions. I have 3 partitions. 1. OS [683 GB]  2. Recovery [14 GB] 3. Ubuntu 12.04 LTS (E:)  I know it says Ubuntu but when I installed Ubuntu I think I put 14 gigabytes for it so I think its the recovery one but I'm not sure. I could have sworn I put more than 700 megabytes for it.  I have googled this and I can't seem to find an answer. Any help would be nice. So here is one I wish to know:   I use Tri-Gate second effect, which allows me to target and banish one card on the field. Then I link that with another monster to bring out another link. Tri-Gate would be in the graveyard but then I use transcodes effect (1 of the 3 co-links) to bring it back from the graveyard.   Can I use the second effect again?