Courant & Robbins Supplement to Chapter 1 "The Theory of Numbers" [p. 21 - 51]
Prime numbers are important because every number may be uniquely expressed as a product of prime numbers. This is noteworthy because prime numbers are a subset of the natural numbers and are initially defined in terms of repeated addition.
Here are a few important facts about prime numbers:
There are infinitely many primes (proof by contradiction, first due to Euclid). This leads to a recursive approach for generating prime numbers (each new prime is the product of the previous primes plus 1.)
Here are the next 5 primes, beginning with 2 and 3.
I generated these using the calculator software on my laptop. One might wonder if they are indeed all primes. For example, did I inadvertently press the wrong key at some point?
Using Mathematica, I can obtain the factors for a number. Here are the results: (web, Mathematica)
Fundamental Teorem of Arithmetic: Every integer N greater than 1 can be factored into a product of primes in only one way.