Thursday, January 29, 2015

An infinitude of primes

Euclid provided a proof for the infinitude of primes.  The following proof is based on his argument:

Theorem 1:  There are an infinite number of primes

(1)  Assume that $p_n$ is the largest prime.
(2)  Let $q = p_n! + 1$
(3)  Since $q > p_n$, $q$ cannot be a prime (by our assumption).
(4)  So, therefore a prime $p$ divides $q$
(5)  But if $p \le p_n$, then $p | p_n!$ and $p | (p_n! + 1) - p_n! = 1$ which is impossible.
(6)  Therefore, we have a contradiction since $p > p_n$ but $p$ is a prime.

QED


Sunday, December 28, 2014

Introduction

The purpose of this blog will be for me to focus on primes and the fundamental theorems related to them.

These will represent my notes.  The goal is to present the fundamental ideas of number theory in as clear a manner as I can without requiring any background beyond standard arithmetic, a genuine interest in math, and an openness to the concepts involved.

The content will be taken straight from number theory text books, classic math papers, and forums on math web sites.

My favorite math web site for asking questions about mathematics is math.stackexchange.com.  If you haven't been to this site, I encourage you to visit if you have any questions.