Prove by complete induction that any natural number $n\ge2$ is equal to a product of prime numbers.
I'm trying to get through Spivak's Calculus on my own and even though I kinda understand induction I'm not so sure that's the case when it comes to complete induction. So I tried to do a starred problem which involves using it. But I'm not sure that my proof is valid, can someone check this for me?  So the problem is that I have to prove that     For every natural $n&gt;1$, if $n$ is not prime we can write it as a product of primes.   I'm starting with a base case. For $n=2$, $n$ is prime, so the assumption holds.  Let's assume that for some $n &gt; 1$ and also all numbers $p &lt;= n$ the assumption holds.  Then $n+1 = cd$. If $n+1$ is not prime, then $c &lt; n+1$ and $d &lt; n+1$, but as they are natural numbers, we can also write that $c &lt;= n$ and $d &lt;= n$. But we assumed that if $p &lt;= n$, and $p$ is not prime, we can write it as a product of primes. So if $c$ or $d$ are not primes, we can write them as a product of primes. That means that $n+1$ can be written as a product of primes if it's not prime, which completes the proof.
Disable white background of pasted text in PowerPoint When I paste text from another source with "Keep source formatting option" into PowerPoint 2010 I receive a white background, that I don't want.    How can I make this text background transparent, so there would be just red text on purple? Rebel flight suits include an ejection harness, Biggs tells Porkins to eject. This seems to suggest that Rebel fighters have an ejection system.  How would they operate for different fighters? Does the whole cockpit separate? Do they function like current aircraft ejection systems - if so what protects the pilot from the vacuum of space? Elastic IP address on EC2 instance interface without 1:1 NAT There is some conflict or error I don't understand. I can skip it and nothing visible happens. The output is okay and everything seems to be fine. Still I would like to get rid of the error message. If I run the following minimal example:   \documentclass[]{scrbook}  \usepackage[applemac, grafiken]{tahbiblatex} \usepackage[ngerman]{babel}   \begin{document}  Testing äüöaüä.  \end{document}   I get the error message:      ! Missing = inserted for \ifnum. &lt;to be read again&gt;                     , l.4 \usepackage                [ngerman]{babel} ?  ! Missing number, treated as zero. &lt;to be read again&gt;                     , l.4 \usepackage                [ngerman]{babel}   Here is my .sty file:  \NeedsTeXFormat{LaTeX2e}\relax \ProvidesPackage{tahbiblatex}[2012, by tah modified by sm]  \RequirePackage{ifthen}  \DeclareOption{latin1}{\PassOptionsToPackage{\CurrentOption}{inputenc}} \DeclareOption{applemac}{\PassOptionsToPackage{\CurrentOption}{inputenc}} \DeclareOption{cp850}{\PassOptionsToPackage{\CurrentOption}{inputenc}} \DeclareOption{utf8}{\PassOptionsToPackage{\CurrentOption}{inputenc}}  \DeclareOption{dvi}{% \newif\ifpdf \ifx\pdfoutput\undefined     \pdffalse \else     \pdfoutput=0     \pdftrue \fi}  \newcounter{Grafiken}\setcounter{Grafiken}{0} \DeclareOption{grafiken}{\setcounter{Grafiken}{1}}  \newcounter{Randausgleich}\setcounter{Randausgleich}{0} \DeclareOption{randausgleich}{\setcounter{Randausgleich}{1}}  \newcounter{Tabellen}\setcounter{Tabellen}{0} \DeclareOption{tabellen}{\setcounter{Tabellen}{1}}  \newcounter{Sync}\setcounter{Sync}{0} \DeclareOption{sync}{\setcounter{Sync}{1}}  \ProcessOptions  \RequirePackage{fixltx2e} \RequirePackage{inputenc} \RequirePackage[T1]{fontenc} \RequirePackage[T1]{url} \RequirePackage{setspace} %\RequirePackage[german]{varioref} \RequirePackage{textcomp}  \newcommand{\urlangabe}[2]{% $\langle$\url{#1}$\rangle$ -- Zugriff am #2}  \def\hierlockern {\begingroup \def\par{\endgraf\endgroup}%   \tolerance 1414%   \hbadness 1414%   \emergencystretch 1.5em%   \hfuzz 0.3pt%   \vfuzz \hfuzz%   \raggedbottom% }  \newcommand{\alleslockern}{%   \tolerance 1414%   \hbadness 1414%   \emergencystretch 1.5em%   \hfuzz 0.3pt%   \vfuzz \hfuzz%   \raggedbottom% } \ifthenelse{\value{Grafiken}=1}   {   \ifx\pdfoutput\undefined \newcount\pdfoutput \fi   \ifcase\pdfoutput       \RequirePackage[dvips]{graphicx}       \DeclareGraphicsExtensions{.eps}       \typeout{Paket tah Meldung: Die Option `grafiken' fuer DVI wurde gewaehlt.}% \else     \RequirePackage[pdftex]{graphicx}     \DeclareGraphicsExtensions{.jpg,.tif,.png,.pdf}     \typeout{Paket tah Meldung: Die Option `grafiken' fuer PDF wurde gewaehlt.}%   \fi}   {}  \ifthenelse{\value{Sync}=1}   {     \ifx\pdfoutput\undefined \newcount\pdfoutput \fi     \ifcase\pdfoutput     \RequirePackage{srcltx}     \typeout{Paket tah Meldung: Die Option `sync' fuer DVI wurde gewaehlt.}%     \else     \RequirePackage{pdfsync}     \typeout{Paket tah Meldung: Die Option `sync' fuer PDF wurde gewaehlt.}%     \fi }   {}   \ifthenelse{\value{Randausgleich}=1}   {\typeout{Paket tah Meldung: Die Option `randausgleich' wurde gewaehlt.}%   \RequirePackage[activate]{pdfcprot}}   {}  \ifthenelse{\value{Tabellen}=1}   {\typeout{Paket tah Meldung: Die Option `tabellen' wurde gewaehlt.}%   \RequirePackage{tabularx}   \RequirePackage{ragged2e}   \RequirePackage{dcolumn}   \RequirePackage{booktabs}   \newcolumntype{L}{&gt;{\RaggedRight\hspace{0pt}\arraybackslash}l}   \newcolumntype{Y}{&gt;{\RaggedRight\hspace{0pt}\arraybackslash}X}   \newcolumntype{d}[1]{&gt;{\DC@{.}{,}{#1}}l&lt;{\DC@end}}   }   {}    \endinput   I also don't really need all the features. So it would be nice to reduce it to minimal features, which are needed.. shell scripts are still working without #!(sha-bang line) I am new to shell scripts and many books have written that use #!(sha-bang) line in starting of the script to invoke the interpreter .And this will invoke a new shell for script and do the interpretation line by line.But one of my basic script is still running without the magic line.  so my questions are:   from where did my basic script get the interpreter. how did the script manages to locate the interpreter.   now let me tell you about my basic script it just contains the following line:  echo "basic script without the magic lines" Personalize monitor position before login I'm using multiple monitors.  When logged in everything is fine: The left monitor is positioned to the left and the right monitor is positioned to the right, but before I log in, the login-screen positions them the other way around.  This is also the position presented by bios and then the grub menu.  I don't mind where the login is displayed, but when moving the mouse, the boundary is the wrong way around and that's annoying. Google AJAX API - How do I get more than 4 Results? Why is "that"/"which" omitted in this sentence? Simulating data to fit a mediation model Carry over long sequence of missing values with Postgres I have a table like this:  create table foo (foo_label text, foo_price int, foo_date date);  insert into foo (           values           ('aaa', 100,  '2017-01-01'),           ('aaa', NULL, '2017-02-01'),           ('aaa', NULL, '2017-03-01'),           ('aaa', NULL, '2017-04-01'),           ('aaa', 140,  '2017-05-01'),           ('aaa', NULL, '2017-06-01'),           ('aaa', 180,  '2017-07-01')         );   As you can see a few values on the foo_price column are missing.  What I need is that missing values get filled up with the "previous" available value in this way:   foo_label | fixed_foo_price | foo_date -----------+-----------------+------------  aaa       | 100             | 2017-01-01  aaa       | 100             | 2017-02-01  aaa       | 100             | 2017-03-01  aaa       | 100             | 2017-04-01  aaa       | 140             | 2017-05-01  aaa       | 140             | 2017-06-01  aaa       | 180             | 2017-07-01   My attempt:  select      foo_label,      (case when foo_price is null then previous_foo_price else foo_price end) as fixed_foo_price,     foo_date from (   select        foo_label,        lag(foo_price) OVER (PARTITION BY foo_label order by foo_date::date) as previous_foo_price,        foo_price,       foo_date       from foo ) T;   As you can see from here:    It doesn't fill completely the '100' series.  Any idea how can I get the wanted result? Just doing this for revision, seems much harder than it should be, should I use $x=\tan u$ ? Any help appreciated. Generating values from a multivariate Gaussian distribution I am currently trying to simulate values of a $N$-dimensional random variable $X$ that has a multivariate normal distribution with mean vector $\mu = (\mu_1,...,\mu_N)^T$ and covariance matrix $S$.  I am hoping to use a procedure similar to the inverse CDF method, meaning that I want to first generate a $N$-dimensional uniform random variable $U$ and then plug that into the inverse CDF of this distribution, so to generate value $X$.  I am having issues because the procedure is not well documented and there are slight differences between the  and a .  In my case, I am also choosing the parameters of the distribution randomly. In particular, I generate each of the means, $\mu_i$, from a uniform distribution $U(20,40)$. I then build the covariance matrix $S$ using the following procedure:   Create a lower triangular matrix $L$ where $L(i,i) = 1$ for $i=1..N$ and $L(i,j)  = U(-1,1)$ for  $i &lt; j$ Let $S = LL^T$ where $L^T$ denotes the transpose of $L$.   This procedure allows me to ensure that $S$ is symmetric and positive definite. It also provides a lower triangular matrix $L$ so that $S = LL^T$, which I believe is required to generate values from the distribution.  Using the guidelines on Wikipedia, I should be able to generate values of $X$ using a $N$-dimensional uniform as follows:   $X = \mu + L * \Phi^{-1}(U)$   According to the MATLAB function however, this is typically done as:   $X = \mu + L^T * \Phi^{-1}(U)$   Where $\Phi^{-1}$ is the inverse CDF of a $N$-dimensional, separable, normal distribution, and the only difference between both methods is simply whether to use $L$ or $L^T$.  Is MATLAB or Wikipedia the way to go? Or are both wrong? How to stop Time Warner cable search on Firefox for Android? I went into about:config and have the following:  browser.search.defaultenginename Google  browser.search.official true  browser.search.order.1 Google  keyword.URL https://www.google.com/search?p=   And still, when I type in the search bar, it pulls up Time Warner Cable Road Runner search instead of Google. It looks like only if I have more than 1 word does it decide to use google -- maybe using only 1 word means I type a malformed URL? How can I fix this to use Google as a proper search engine? How to break the Y axis in an Excel sheet I want to break the Y-axis in an Excel sheet. How can I do that?  For example, suppose I have the data in a range from 0-7000 and then range from 22000-28000. I want a break in the Y axis from 8000-20000. What to call the attitude of pursuing something Laplace equation of composite function Suppose we have $f(x,y)$, which satisfies Laplaces equation: $\frac{\delta^2f}{\delta x^2}+\frac{\delta^2f}{\delta y^2}=0$  Does $f(\frac{x}{x^2+y^2},\frac{y}{x^2+y^2})$ also satisfy Laplace equation? What is the difference between \em and \emph? What is the difference between \emph{...} and {\em ...}?  When to use each of them? Alternative definitions of wifebeater I was a li'l nonplussed to find out that the word  can mean other things besides "a man who beats up his wife". Its definition reads:        One who (usually as a repeated practice) beats one’s wife, or a husband prone to violence   A kind of sleeveless shirt, often but not exclusively worn as an undershirt.   (uncountable, UK, slang) Stella Artois, a brand of lager beer.       reckons that sense 2 is American and originates "apparently from the association of such a garment with men who commit domestic violence". It does not register the UK slang. I'm guessing that Stellas also have an unfortunate association with domestic violence.  Are these alternative senses popular in their respective countries or are they perhaps nonce words that have lingered on? Any idea how these disparaging associations came about? I expect that the brewers of Stella Artois are none too happy about it. jQuery document.createElement equivalent? I am having some issues with the hyperref package. I would like to create a Introduction chapter without a number so I use \chapter*{Introduction} and \addcontentsline{toc}{chapter}{Introduction} because I want it in the table of contents (see example).  But then the link in my PDF viewer gets it wrong and also the automatically created header is no longer correct (see picture). The table of contents shows the right page number (p.1) but the link is also wrong.   What is my mistake here?  \documentclass{scrbook}  \usepackage{hyperref} \usepackage{blindtext} \usepackage{glossaries}   \makeglossaries  \newacronym{eg}{e.g.}{% for example }   \begin{document}   \frontmatter  % abstract before the table of contents \chapter*{Abstract} \addcontentsline{toc}{chapter}{Abstract} \blindtext  \tableofcontents  % chapter with acronyms \cleardoublepage \phantomsection  \printglossary[type=\acronymtype] \addcontentsline{toc}{chapter}{Glossary}  \cleardoublepage \mainmatter   \phantomsection % \chapter{Introduction} % with this everything is correct but I don't want the number \chapter*{Introduction} \addcontentsline{toc}{chapter}{Introduction}  \blindtext  \pagebreak  \blindtext  % first chapter \chapter{Chapter1}  \gls{eg} \blindtext  \pagebreak  \blindtext   \end{document}