How can I prove $5^{3^n}+1$ is divisible by $3^{n+1}$
Prove by mathematical induction that $2^{3^n}+1$ is divisible by $3^{n+1}$
Korea: Visa required to go to the city while in transit in Seoul for a Philippine citizen? Hi I'm traveling to Uzbekistan and have 12 hours layover in Seoul via Korean Airlines. Do I need a visa to go out of the airport for sightseeing? I'm a Philippine passport holder. Do I have to worry about copyright issues for code posted on Stack Overflow? I was wondering about the copyright of code snippets from answers. Let's say you ask a question and the solution consists of ten lines of code, which are given in an answer. Are there any restrictions about the use of that code?   It seems that the content of this page is under the . So does one have to attribute the author if code is used in an open source project? Can it be legally used in closed source projects at all?  I guess ten lines of code are not really problematic, but is there a critical number of lines? I am reading Boyd &amp; Vandenberghe's Convex Optimization. The authors use curved greater than or equal to (\succcurlyeq)  $$f(x^*) \succcurlyeq \alpha$$  and curved less than or equal to (\preccurlyeq)  $$f(x^*) \preccurlyeq \alpha$$  Can someone explain what they mean? Choosing my name as an author when publishing a scientific paper, can I use my "unofficial" first name? I have one official first name which appears on my ID card. I have another first name which friends call me by. Is it possible to use my unofficial first name and official family name to publish scientific papers? How can I prove to someone that I am one of the authors? C++ compiling on Windows and Linux: ifdef switch I am stuck while calculating the peak current in the boost converter through the inductor.  Can you please check if I am right?  Ipeak=(Max Voltage/ R2) = 18/1000=0.018. So, my peak current through the inductor is just 18mA? Am I doing it correct. If this is correct, how will I account for the Vce drop inside the pin1 and pin2 of the IC. The darlington transistor inside the IC is just like a normal transistor. I have added two transistor outside the IC instead of using the darlington switch in the IC for thermal purposes.   Can you please help me with the calculation of Vswce and Ipeak. The calculation of Vswce is not mentioned in the datasheet. So, please help. And the calculation of Peak current is mentioned in the datasheet. But is my approach of calculating the peak current through the inductor right? AJAX select returns 'illegal choice detected' Create a settings form for a module in Drupal 8 - when a user selects a View from a list, the drop down under it should populate with the displays in that view.  While the form updates the option list for the user, it also throws an error message on the screen "Illegal choice has been detected".  Here is the code (it doesn't paste very well):    getViewDisplays is the ajax callback that is populating the options.  Some people who have posted about this around the internet say to set #validated on the select field to TRUE, though, that doesn't necessarily feel right. I still want to validate the submission to do some checks. Unfortunately, I need to do this.  I'm using ELMAH for my error log.  Before I route to my error.aspx view, I have to grab the default ELMAH error log so I can log the exception.  You used to be able to use   Elmah.ErrorLog.Default   However, this is now marked as obsolete.  The compiler directs me to use the method  Elmah.ErrorLog.GetDefault(HttpContext context)   MVC's context is of type HttpContextBase, which enables us to mock it (YAY!).  How can we deal with MVC-unaware libraries that require the old style HttpContext? Is there an advantage to use a Synchronized Method instead of a Synchronized Block? How to get Microsoft Search to index common file types like .pdf and .pub I'm trying to get Windows Search (4.0) to index PDFs and .pub, .doc, etc. files.  The files are on an XP Pro PC acting as a file server. But I don't have MS Office, etc. installed (no need, trying to keep it "slim and trim") but WS 4.0 is doing the indexing of those files and not indexing .pdf and .pub.  I've found a few  (from Foxit PDF reader) rated 3/5 and only support WS 2.0 or 3.0.  Looking for other solutions and also a way to index .pub files. Detect Chrome extension first run / update What is a Pythonic way to pad a numeric string with zeroes to the left, i.e. so the numeric string has a specific length? Move EFI System Partition to another drive If I have an adjacency matrix for a graph, can I do a series of matrix operations on the adjacency matrix to find the connected components of the graph? I would like to transfer files between two remote hosts using on local shell, but it seems rsync doesn't support synchronisation if two remotes are specified as follow:  $ rsync -vuar host1:/var/www host2:/var/www The source and destination cannot both be remote.   What other workarounds/commands I could use to achieve similar results? 80's pop song with lyric "It's the story of my life if you wanna know" I'm trying to identify a song by a female 80's pop group.   Here are the lyrics I remember:     It's the story of my life if you want to know   a play where everything is meant to be   you're the story of my life and I can't let you go When does $\,n\mid q^3 \Rightarrow n\mid q\,$? I think this is true because of prime factorisations, i.e.  If $3$ a factor of the prime factorisation of $q^3$, then $3$ is a factor of the prime factorisation of $q$.  Therefore If $3$ divides $q^3$, then $3$ divides $q$.  Or for a general case, if $n$ divides $q^3$; then $n$ divides $q$. Is this correct? Let $f$ be a bounded measurable function on a set of finite measure $E$.  For a measurable subset $A$ of $E$, show that $\int_A f = \int_E f \cdot \chi_A$     Lemma: Let $f$ be a bounded measurable function on a set of finite measure $E$.  Suppose $A$ and $B$ are disjoint measurable subsets of $E$.  Then $\int_{A \cup B} f = \int_A f + \int_B f$.   My first approach was to view the problem like this utilizing the above lemma:  $$\int_E  f \cdot \chi_A = \int_{E\setminus A} f \cdot X_A + \int_A f \cdot \chi_A $$  I'm not really sure how the result follows from the above. Every separable Banach space is isomorphic to $\ell_1/A$ for some closed $A\subset \ell_1$ How to prove the following mind-blowing fact?     Let $X$ be a separable Banach space and let $\ell_1$ be the space of all absolutely summable scalar sequences. Then there exists such closed subspace $A\subset \ell_1$ that factor space $\ell_1/A$ and $X$ are isomorphic as normed spaces.   Edit: So what, this is like a classification up to isomorphism of all separable Banach spaces? Each separable Banach space corresponds to some closed subspace of $\ell_1$? I'm using Drupal 7.  Is it possible to add a CC to a webform email?  I was using   function theme_webform_mail_headers($variables) {   $headers = array(     'X-Mailer' =&gt; 'Drupal Webform (PHP/' . phpversion() . ')',   );   return $headers; }   changing it with      function MYTHEME_webform_mail_headers($variables) {       $headers = array( 'CC' =&gt; 'ccmeail@domain.com', 'X-Mailer' =&gt; 'Drupal Webform (PHP/'. phpversion() .')',  );       return $headers;     }   It does not work, I've cleared all the chaches and it does not add the CC. How could I add the CC?  Thanks in advance.