ahafidi

Irrationality of Square Roots of Prime Numbers

We all remember from our fundamental mathematics classes the emphasis on the irrationality of $\sqrt{2}$ using a simple proof by contradiction.

Well, it turns out that we can extend this theorem more broadly to the square roots of prime numbers with the same kind of demonstration, and that's what we're going to demonstrate here in this article. But first, let's have some reminders about what prime numbers are.

Let $p$ be a natural number, we say that $p$ is prime if and only if $p$ is different from 1 and it has only 1 and itself as divisors. Let's denote this set of prime numbers as $\mathbb{P}$.

$$\forall p \in \mathbb{N}, \ p \in \mathbb{P} \iff \begin{cases} p \neq 1 \newline \nexists(a, b) \in \mathbb{N}^2/ \ a \neq 1 \And b \neq p \ \Longrightarrow \ p = ab \end{cases}$$

steven2358

FFmpeg cheat sheet

A list of useful commands for the FFmpeg command line tool.

Download FFmpeg: https://www.ffmpeg.org/download.html

Full documentation: https://www.ffmpeg.org/ffmpeg.html

Basic conversion

beanmoss

<?php namespace App\Http\Controllers;

use Illuminate\Http\Request;

trait RestControllerTrait
{
	public function index()
    {
        $m = self::MODEL;
        return $this->listResponse($m::all());

asika32764

/<\?php\s(\/\*(?:[^*]|\n|(?:\*(?:[^\/]|\n)))*\*\/)\s/

Test in: http://www.rubular.com/

isaacs

echo 'export PATH=$HOME/local/bin:$PATH' >> ~/.bashrc
. ~/.bashrc
mkdir ~/local
mkdir ~/node-latest-install
cd ~/node-latest-install
curl http://nodejs.org/dist/node-latest.tar.gz | tar xz --strip-components=1
./configure --prefix=~/local
make install # ok, fine, this step probably takes more than 30 seconds...
curl https://www.npmjs.org/install.sh | sh