Let $ a, b \in\mathbb {Z} ^ + $ such that gcd$ (a, b) = 1 $. If $ ab = z ^ n $ for some $ n \in \mathbb {N} $ and $ z \in \mathbb {Z} $, then there are $ x, y \in \mathbb {Z} $ such that $ a = x ^ n $ and $ b = y ^ n $.
Show that if $n$, $a$, $b$, and $c$ are positive integers with $\gcd(a, b) = 1$ and $ab = c^n$, then there are positive integers $d$, and $e$ such that $a = d^n$ and $b = e^n$.   I know that (by Bezout) $\gcd\left(a,b\right) = 1$ implies $ax + by = 1$ for some integers $x$ and $y$, and also that $\gcd\left(a^n,b^n\right) = 1$, but this does not help me.
Weird http -> https redirection although not configured I am running Apache 2.2.15 (CentOS 6.6) with a HTTP-only domain   demo.xml-director.info  Using wget I can retrieve the content properly   wget -S http://demo.xml-director.info --2015-01-05 13:31:41--  http://demo.xml-director.info/ Resolving demo.xml-director.info (demo.xml-director.info)... 176.9.146.28 Connecting to demo.xml-director.info (demo.xml-director.info)|176.9.146.28|:80... connected. HTTP request sent, awaiting response...    HTTP/1.1 200 OK   Date: Mon, 05 Jan 2015 12:31:41 GMT   Server: Zope/(2.13.22, python 2.7.6, linux2) ZServer/1.1   Content-Length: 20227   Expires: Sat, 01 Jan 2000 00:00:00 GMT   Content-Type: text/html;charset=utf-8   X-Ua-Compatible: IE=edge,chrome=1   Content-Language: en   Vary: Accept-Encoding   Keep-Alive: timeout=15, max=100   Connection: Keep-Alive Length: 20227 (20K) [text/html] Saving to: 'index.html.4'   However Chrome/Firefox always change the request from http to https because of HTST. However for this particular domain there is no HSTS configured.  The server runs SSL for www.xml-director.info with enabled HSTS support. However there is further alias here that would map demo.xml-director.info to www.xml-director.info.   How can this problem be resolved.  VHOst for www.xml-director.info:  &lt;VirtualHost *:443&gt;     ServerName www.xml-director.info     ServerAlias xml-director.info      SSLEngine on     SSLCertificateFile /etc/httpd/certs/15742445repl_2.crt     SSLCertificateCHainFile /etc/httpd/certs/15742445repl_2.ca-bundle     SSLCertificateKeyFile /etc/httpd/certs/zopyx.com.key      CustomLog /var/log/httpd/xml-director.info.log combined     DocumentRoot /var/www/xml-director/landing-v1      Header always set Strict-Transport-Security "max-age=63072000; includeSubDomains"      &lt;location "/"&gt;         Options +Indexes         Header set Access-Control-Allow-Origin "*"         Header set Access-Control-Allow-Methods "GET,PUT,POST,DELETE"     Header set Access-Control-Allow-Headers "X-Requested-With,Content-Type"     &lt;/location&gt;  &lt;/VirtualHost&gt;  &lt;VirtualHost *:80&gt;     ServerAlias www.xml-director.com      ServerAlias www.xml-director.info     ServerAlias www.xml-director.de     ServerAlias xml-director.com     ServerAlias xml-director.info     ServerAlias xml-director.de      RedirectPermanent / https://xml-director.info/ &lt;/VirtualHost&gt;   and for demo.xml-director.info  &lt;VirtualHost *:80&gt;     ServerName  demo.xml-director.info     RewriteEngine On      RewriteRule ^/(.*) http://127.0.0.1:12020/VirtualHostBase/http/demo.xml-director.info:80/xml-director/VirtualHostRoot/$1 [L,P]     RewriteRule ^/$    http://127.0.0.1:12020/VirtualHostBase/http/demo.xml-director.info:80/xml-director/VirtualHostRoot/$1 [L,P]      CustomLog /var/log/httpd/demo.xml-director.info.log combined     AddOutputFilterByType DEFLATE text/html text/xml text/plain text/css text/javascript      CacheRoot /tmp/cache     CacheEnable disk /     CacheIgnoreCacheControl on      KeepAliveTimeout 15     KeepAlive on      ExpiresActive On     ExpiresByType image/gif "access plus 10 day"     ExpiresByType image/jpg "access plus 10 day"     ExpiresByType image/jpeg "access plus 10 day"     ExpiresByType image/png "access plus 10 day"     ExpiresByType text/javascript "access plus 10 day"     ExpiresByType text/html  "access plus 1 hour"     ExpiresByType text/html  "access plus 1 hour"     ExpiresByType text/css  "access plus 10 days"      &lt;Location "/"&gt;         Order allow,deny         Allow from all     &lt;/Location&gt;  &lt;/VirtualHost&gt; If it is given that component of acceleration perpendicular to the velocity of a body has a constant, non-zero magnitude, how can we mathematically prove that the trajectory of the body will be circular?   I have basic knowledge of calculus, vectors and co-ordinate geometry. I can prove that the trajectory of a body in projectile motion is parabolic. I tried to follow a similar method to get an equation of circle but couldn't succeed. I tried forming parametric equations of x and y in terms of time t (the way we proceed when proving the parabolic path of a projectile) but could not reach the conclusion.  Edit: I have edited the question in order to improve it. However I still can not share my work on this question as I had asked the question almost 6 months ago and now I can not find where I tried solving it. Sincerely sorry for that. I do not know how to prove this question: $m^{\phi(n)}+n^{\phi(m)}\equiv 1 \pmod {mn}$ where $m$ and $n$ are relatively prime.  Can anyone help? On some user profiles, e.g.  image proportions are broken:        Image on activity tab or on  page isn’t distorted:        The issue is probably related to the old one: What is the quantum mechanical explanation of the octet rule? What is the quantum mechanical explanation of the ? In other words, what makes the octet rule be true from a quantum mechanical view? How we explain  what makes some atoms don't follow the octet rule using QM? Including null values in unpivoted column Cylindrical vs azimuthal symmetry I'm working on a WordPress site for my band and I'd like to mark every 3rd post on our blog page to have a special class applied to it, anybody have any pointers on how to achieve this? Any help is very very appreciated, thanks! rock n roll. How to get directory size in PHP I think I'm being the victim of a bug here. Sometimes while I'm working (I still don't know why), my network traffic goes up to 200 KB/s and stays that way, even tough I'm not doing anything internet-related.   This sometimes happens to me with the CPU usage. When it does, I just run a top command to find out which process is responsible and then kill it. Problem is: I have no way of knowing which process is responsible for my high network usage. Both the resource monitor and the top command only tell me my total network usage, neither of them tells me process specific network info.  I've found questions here about monitoring total bandwidth usage, but, as I mentioned, that's not what I need. Is there another command I can use to find out which process is getting out of hand?  The command iftop gives results that disagree entirely with the information reported by System Monitor. While the latter claims there's high network traffic, the former claims there's barely 1 KB/s.  I've already tried killing all the obvious ones (Firefox, update-manager, Pidgin, etc) with no luck. So far, restarting the machine is the only way I found of getting rid of the issue. Apply a string replacement on the citation as returned by \cite How to call Apex REST API from Javascript? I have two different developer orgs. One contains apex REST web service, and I want to call that REST service from my second org. I have tried to call that REST service using normal ajax post request and forcetk.js, But I got following error     XMLHttpRequest cannot load   . Origin    is not allowed by   Access-Control-Allow-Origin.   I searched about this problem then I got to know that Salesforce never allows CORS (Cross-Origin-Resource-Sharing).  Initially I was using apex to call that REST web service, that time I was able to post data from apex. But as we have callout limitation in salesforce, I want to use javascript.  Can anybody please help me? Different PDF output with same source files (NeurIPS 2021 package) Decreasing font size in verbatim How can I modify the following command to make it work?  \newcommand{\code}[1]{\begingroup \fontsize{10pt}{10pt}\selectfont \begin{verbatim} #1 \end{verbatim} \endgroup}   I currently get the following error on the close of the following verbatim block:  ! Undefined control sequence. &lt;argument&gt; ...endgroup ^^M^^M$\eta_{xi} = r_{xi} -                                                    q_i \cdot p_x$ \\^^M$q_i ... l.227     \end{verbatim} For some reason I can't unhide my objects anymore in this file. Is it a bug? In MLE for continuous rv, why is it ok to evaluate a pdf at a point? What does = +_ mean in JavaScript Degree of minimum polynomial at most $n$ without Cayley-Hamilton? Derivation of Frey equation from FLT I understand, on  a layman's level, Fray's motivation to write an elliptic equation corresponding to an assumed solution to FLT. My question is, how technically is Frey's equation derived?   $1.$ FLT : Contradiction. There exist integers $A,B,C,n&gt;2$ such that $A^n + B^n = C^n$  $2.$  Frey: $y^2 = x(x-A^n)(x+B^n)$  $3.$  Elliptic curve $y^2= x^3 + ax + b$    $4.$  The curve is not modular. => not elliptic => FLT  proved.   I need is the algebra to go from 1. to 3. above.  Assume Numbers A,B,C such that A^n + B^n = C^n contradicting FLT.  $y^2 = x^3 + ax + b$ an elliptic curve in Weirstrass form. I am guessing A and B go unchanged into x(x-A^n)(x+B^n)   Is that correct ?  Expanding 2. I get a term in x^2 , there is none in 3. This is my problem.  What does a $2$ dimensional graph of the Frey curve look like ? Can QGIS export Geospatial PDF for Avenza PDF map app? Since QGIS uses GDAL, does any map exported as a PDF become a Geospatial PDF? I am hoping to import maps from QGIS into the Avenza PDF Map app on my phone. Previous posts mentioned exporting as Geospatial PDF's and GeoTiff's, but none had an exact answer or process.