I made a program to find out the number of primes within a certain range, for example between $1$ and $10000$ I found $1229$ primes, I then increased my range to $20000$ and then I found $2262$ primes, after doing it for $1$ to $30000$, I found $3245$ primes.  Now a curious thing to notice is that each time, The probability of finding a prime in between $2$ multiples of $10000$ is decreasing, i.e it was $$\frac{2262-1229}{10000}=0.1033$$ between $10000$ and $20000$, and $$\frac{3245-2262}{10000}=0.0983$$ between $20000$ and $30000$,   So from this can we infer that there will exist two numbers separated by a gap of $10000$ such that no number in between them is prime? If so how to determine the first two numbers with which this happens? Also I took $10000$ just as a reference here, what about if the gap between them in general is $x$, can we do something for this in generality?  Thanks!
Hello and I'm quite new to Math SE.  I am trying to find the largest consecutive sequence of composite numbers. The largest I know is:  $$90, 91, 92, 93, 94, 95, 96$$  I can't make this series any longer because $97$ is prime unfortunately.  I can however, see a certain relation, if suppose we take the numbers like (let $a_1, a_2, a_3,...,a_n$denote digits and not multiplication):  $$a_1a_2a_3...a_n1,\ a_1a_2a_3...a_n2,\ a_1a_2a_3...a_n3,\ a_1a_2a_3...a_n4,\ a_1a_2a_3...a_n5,\ a_1a_2a_3...a_n6,\ a_1a_2a_3...a_n7,\ a_1a_2a_3...a_n8,\ a_1a_2a_3...a_n9,\ a_1a_2a_3...(a_n+1)0$$  The entire list of consecutive natural numbers I showed above can be made composite if:   The number formed by digits $a_1a_2a_3...a_n$ should be a multiple of 3 The numbers $a_1a_2a_3...a_n1$ and $a_1a_2a_3...a_n7$ should be composite numbers   If I didn't clearly convey what I'm trying to say, I mean like, say I want the two numbers (eg: ($121$, $127$) or ($151$, $157$) or ($181$, $187$)) to be both composite.  I'm still quite not equipped with enough knowledge to identify if a random large number is prime or not, so I believe you guys at Math SE can help me out.
Assume that $Y_0,\ldots , Y_n$ are independent random variables with the following identical distribution: $Y_i=1$ with propability $p$ and $Y_i=0$ with propability $1-p$. Also set $S_0=0$ and $S_n=Y_0+\ldots + Y_n$.  It's quite trivial that $(Y_n)_{n\geq 0}$ is a Markov chain, and I managed to show that $(X_n)=(S_n)_{n\geq 0}$ is also a Markov chain, by showing that  $\mathbb{P}(X_{n+1}=i_{n+1}\,|\,X_0=i_0,\ldots,X_n=i_n)=\mathbb{P}(X_{n+1}-X_n=i_{n+1}-X_n\,|\,X_0=i_0,\ldots,X_n=i_n)=\mathbb{P}(Y_{n+1}=i_{n+1}-X_n\,|\,X_0=i_0,\ldots,X_n=i_n)=\mathbb{P}(Y_{n+1}=i_{n+1}-i_n)=\mathbb{P}(Y_{n+1}+i_n=i_{n+1})=\mathbb{P}(X_{n+1}=i_{n+1}\,|\,X_n=i_n)$  However I am now running into the problem that I have to find whether or not $(X_n)=(S_0+\ldots + S_n)_{n\geq 0}$ is a Markov chain. I suspect that it isn't, but i'm having trouble finding a counterexample. I also need to find if $(X_n)=(S_n, S_0+\ldots+S_n)$ is a Markov chain, but I think this actually might be one, since $X_n$ gives "more information" (so to speak) to $X_(n+1)$. Again, having trouble actually proving that.  I appreciate any help, tips etc you can give me. How do I identify which Linux distro is running? We have some new hardware in our office which runs its own customized Linux OS.   How do I go about figuring which distro it's based on? Is there a GUI IDE for programming python just like java has netbeans?  I wanted one where we could drag &amp; drop test boxes &amp; labels just as we do in swing components using netbeans or eclipse.  Or is there an add-on present in either netbeans or eclipse itself? \textcite macro produces "Author (year)" correctly but colours first paren with hyperref I am using a custom \textcite macro that I found somewhere. It works nicely except that it makes the opening paren of "Author (year)" part of the link (also colouring it blue) which is not desired.  This is likely just a minor change to the definition of the macro, but the solution eludes me (I don't get where the opening paren is added at all).    Opening paranthesis should not be part of link and should not be coloured.  (M)WE:  \documentclass{scrartcl}  \usepackage{csquotes}  \usepackage[backend=biber,             bibencoding=utf8,             hyperref=true,             natbib=true,             minnames=1,     %% prevent truncation to year-only             backref=true, %% TODO REMOVE before final             bibstyle=apa,             citestyle=authoryear,             ]{biblatex} \DeclareLanguageMapping{english}{english-apa} \addbibresource{bib.bib}   \DeclareCiteCommand{\textcite}   {\boolfalse{cbx:parens}}   {\usebibmacro{citeindex}%    \printtext[bibhyperref]{\usebibmacro{textcite}}}   {\ifbool{cbx:parens}      {\bibcloseparen\global\boolfalse{cbx:parens}}      {}%    \multicitedelim}   {\usebibmacro{textcite:postnote}}   \usepackage[table,svgnames]{xcolor}  \definecolor{solblue}{HTML}{268BD2}  \usepackage[pdftex]{hyperref}  \hypersetup{    pdfhighlight = /P,    colorlinks=true,    urlcolor=solblue,    filecolor=solblue,    linkcolor=solblue,    citecolor=solblue,    raiselinks=true,    breaklinks,    unicode }   \begin{document}  foobar \textcite{thesis_pockrandt} foobar  \end{document}   bib.bib:  @phdthesis{thesis_pockrandt,     author = {Christopher Maximilian Pockrandt},     title = {{Approximate String Matching: Improving Data Structures and Algorithms}},     year = {2019} } I am very new to the TikZ package, and I need to draw some flow chars by using the following shapes.   However I couldn't figure out how to draw the 4th and and 6th ones below. It would be very nice (for completeness) to help with all of the shapes in the list below.      Your help will be greatly appreciated.  Note: I have visited  but it contains some other shapes. X Windows special programs: do some of these still exist? Highlighting the selected tab using jQueryUI Tabs I have a header page with links to all the visualforce page as tabs.I need to highlight the tab on select of it.      &lt;apex:page title="{!$Label.site.site_login}" showHeader="false" sidebar="false"       standardStylesheets="false"&gt; &lt;apex:includeScript value="{!URLFOR($Resource.jQuery, '/js/jquery-1.4.2.min.js')}"  /&gt;   &lt;apex:includeScript value="{!URLFOR($Resource.jQuery, '/js/jquery-ui-1.8.6.custom.min.js')}"  /&gt; &lt;apex:stylesheet value="{!URLFOR($Resource.jQuery, '/css/ui-lightness/jquery-ui-1.8.6.custom.css')}"  /&gt;          &lt;html lang="en"&gt;  &lt;head&gt;   &lt;script src="http://ajax.googleapis.com/ajax/libs/jqueryui/1.8/jquery-ui.min.js"&gt;  $(function()  {   $( "#tabs" ).tabs();   });  &lt;/script&gt;   &lt;style&gt;   #tabs .ui-tabs-active    { background: red; }  &lt;/style&gt;     &lt;/head&gt;      &lt;body&gt;         &lt;nav class="navbar navbar-static-top navbar-inverse" role="navigation"&gt;             &lt;div class="container"&gt;                 &lt;div class="navbar-header"&gt;                     &lt;button type="button" class="navbar-toggle" data-toggle="collapse" data-target=".navbar-ex1-collapse"&gt;                         &lt;span class="icon-bar"&gt;&lt;/span&gt;&lt;span class="icon-bar"&gt;&lt;/span&gt;                         &lt;span class="icon-bar"&gt;&lt;/span&gt;                      &lt;/button&gt;                     &lt;a class="navbar-brand" style="line-height: 0px; padding: 10px 15px;" href="index.php.html"&gt;                         &lt;img height="30px" src="{!URLFOR($Resource.Resources, 'img/logo.png ')}" /&gt;                     &lt;/a&gt;                 &lt;/div&gt;                 &lt;div class="collapse navbar-collapse navbar-ex1-collapse" id="tabs"&gt;                     &lt;ul class="nav navbar-nav"&gt;                         &lt;li id="Home"&gt;                             &lt;apex:outputLink value="{!$Page.Home}"&gt;Home&lt;/apex:outputLink&gt;                         &lt;/li&gt;                         &lt;li id="chatter"&gt;                             &lt;apex:outputLink value="{!$Page.Chatter}"&gt;Chatter&lt;/apex:outputLink&gt;                         &lt;/li&gt;                         &lt;li id="page3"&gt;                             &lt;apex:outputLink value="{!$Page.page3}"&gt;Room&lt;/apex:outputLink&gt;                         &lt;/li&gt;                         &lt;li id="page4"&gt;                             &lt;apex:outputLink value="{!$Page.page4}"&gt;Page4&lt;/apex:outputLink&gt;                         &lt;/li&gt;                     &lt;/ul&gt;                 &lt;/div&gt;             &lt;/div&gt;         &lt;/nav&gt;     &lt;/body&gt;      &lt;/html&gt; &lt;/apex:page&gt; When I use any variation of English, US international (with dead keys,altGr dead keys or alternative) on my Linux Mint machine I always encounter this behaviour. When I press one of these keys: ' " and then follow them with a 'non-accentable' character like a [ or a b no output comes out at all.    Whereas in Windows US-International it would print [ or b. If I wanted to type this I would have to escape each dead key with a space instead of with any 'non-accentable' character. This is annoying when programming (not really, but I trained with the Windows 'Qwerty International' on typing.io and switching back and forth between the systems is irritating).    Is there any way to change that so it works like in Windows? Deriving the speed of the propagation of a change in the Electromagnetic Field from Maxwell's Equations I've been told that, from Maxwell's equations, one can find that the propagation of change in the Electromagnetic Field travels at a speed $\frac{1}{\sqrt{\mu_0 \epsilon_0}}$ (the values of which can be empirically found, and, when plugged into the expression, yield the empirically found speed of light)  I'm really not sure how I would go about finding $v = \frac{1}{\sqrt{\mu_0 \epsilon_0}}$ simply from Maxwell's equations in the following form, in SI units --  $$\nabla \cdot \mathbf{E} = \frac {\rho} {\epsilon_0}$$  $$\nabla \cdot \mathbf{B} = 0$$  $$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}$$  $$\nabla \times \mathbf{B} = \mu_0 \left( \mathbf{J} + \epsilon_0 \frac{\partial \mathbf{E}} {\partial t} \right)$$  Is what I believe true? (that the speed of propagation is derivable from Maxwell's Equations)  If not, what else is needed?  If so, can you provide a clear and cogent derivation? Is there a single word or very short phrase to describe the act of contemptuously referring to someone by using a diminutive "nickname", with which that person is not normally associated?  For example, referring to Henry Wadsworth Longfellow as Hank Longfellow? Or William Penn as Billy Penn? The line numbers for the text of a paragraph with an equation defined by \begin{align} command do not show as expected. How to fix this?    The LaTeX codes are attached here.   \documentclass[a4paper, 12pt]{article} \usepackage[utf8]{inputenc}    % Use the UTF-8 encoding \usepackage{amsmath}           % For the use of math \usepackage{amsfonts} \usepackage{amssymb}           % For the use of AMS symbols \usepackage{indentfirst}       % For indentation \usepackage[top=2cm, bottom=2cm, left=2cm, right=2cm]{geometry}  \usepackage{lineno} \linenumbers  \usepackage{titlesec} \titleformat*{\section}{\large\bfseries} \titleformat*{\subsection}{\normalsize\bfseries} \titleformat*{\subsubsection}{\normalsize}   \renewcommand{\baselinestretch}{1.5}    \usepackage{authblk}           % Use author package \renewcommand*{\Authfont}{\large} \renewcommand*{\Affilfont}{\normalsize\normalfont}  \makeatletter \g@addto@macro\normalsize{%   \setlength\abovedisplayskip{0pt}   \setlength\belowdisplayskip{0pt}   \setlength\abovedisplayshortskip{0pt}   \setlength\belowdisplayshortskip{0pt} } \makeatother  \title{\textbf{Globally cumstomize the spacing between the text and equation}}    \date{}  \begin{document}  \maketitle  \section{Air filter} The air filter is simulated by the Control Valve module (AFT in Fig. 2(a)). The pressure drop through the air filter ($ \Delta P_{af} $) is given by the following equation [11,16]. \begin{align} \Delta P = \Delta P_{d} \left( \frac{m}{m_{d}}\right)^{1.84}  \left( \frac{T}{T_{d}}\right)  \left( \frac{P}{P_{d}}\right)^{-1}  \end{align} where, $ \Delta P_{af} $ is the pressure drop, $ m $ is the mass flow rate, $ T $ is the temperature, and $ P $ is the pressure, and subscript $ d $ denotes the design condition.  \newpage \section{Air compressor (AC)} The AC operating characteristics can be described by its performance map, which relates the following dimensionless operational variables. The AC operating characteristics can be described by its performance map, which relates the following dimensionless operational variables. \begin{align} &amp; \text{Relative pressure ratio:} &amp;&amp; PR_{r} = \frac{PR-1}{PR_{d}-1}\\ &amp; \text{Relative isentropic efficiency:} &amp;&amp; \eta_{r} = \frac{\eta}{\eta_{d}} \\ &amp; \text{Relative corrected mass flow:} &amp;&amp; m_{cor,r} = \frac{\left(\frac{m_{in}}{P_{in}}\sqrt{T_{in}}\right)}{\left(\frac{m_{in,d}}{P_{in,d}}\sqrt{T_{in,d}}\right)} \\ &amp; \text{Relative corrected speed:} &amp;&amp; N_{cor,r} = \frac{\left(\frac{N}{\sqrt{T_{in}}}\right)}{\left(\frac{N_{d}}{\sqrt{T_{in,d}}}\right)} \end{align} where, $ \eta $ is the efficiency, $ N $ is the shaft speed, and $ PR = P_{in}/P_{out} $. Subscript $ cor $ denotes the corrected value, $ r $ denotes the relative value, $ in $ denotes the inlet, and $ out $ denotes the outlet.  \end{document} Geomagnetic reversal? Which files shouldn't be checked in into version control in Android Studio How do I get my Python program to sleep for 50 milliseconds? Integrating the inverse of a function. $f(x)$ is a strictly increasing, and continuous, function on $[0,+\infty)$. Consider the integral,  $$\int_0^a f^{-1}(x) dx.$$  Reasoning  The interval $[0,a]$ is "reflected" across the line $y=x$ onto the interval $0 \leq y \leq a.$  $f(x)$ and $f^{-1}(x)$ are symmetric across the line $y=x$. So, the integral is equivalent to the area bounded by $y = 0$, $y=a$, and $f(x)$.  So, "integrating $f$ along an interval on the $y$-axis" is the same as integrating $f^{-1}$ along the same interval on the $x$-axis?  Question  Is my reasoning correct? Is there a name for this type of sentence structure: "She looks as though she's been poured into her clothes, and forgot to say 'when'"? Comedians seem to use phrases that employ this type of sentence structure - is there a name for it?  Examples of Groucho Marx's one liners seem to fit this pattern — and if memory serves, Emo Philips.       One morning I shot an elephant in my pajamas, how he got in my pajamas, I don't know.     I've had a perfectly wonderful evening, but this wasn't it. Electronic visa having Old Passport number How to make a domain receiving email behind cloudflare? I plan to attach my 2014 and 2015 annual leave letters to prove I was on leave, and a letter from my sales manager who was in charge of the business during my vacation.  I intend to explain all my sources of income in additional letter because the ECO calculations of my income were different from mine.   How else can I improve my new application? Remove preinstalled apps on Big Sur MacBook Pro M1 Sorry for my English, I'm French and  I have  tried a lot (even with SIP disabled) but nothing works. I'm kind very desperate since I’m not sure how I would set up screen time to limit my children’s usage of the Pre-installed apps. The reason why I need to do so, is that my kid waste time in stead of studying. Is there a guide how to set this up?