Discrete Mathematical Structures, Lecture 3.4: Divisibility and primes

Discrete Mathematical Structures, Lecture 3.4: Divisibility and primes.

We say that an integer d divides n if n=dk for some integer k. A integer p greater than 1 is prime if p=ab implies either p=a or p=b. We prove that this is equivalent to the more classic "grade school definition" of p having only two (positive) divisors. We show that divisibility is transitive, and how this means that every positive integer is divisible by a prime. We state, without proof, the fundamental theorem of arithmetic, which says that every integer greater than 1 has a unique prime factorization. Though this seems obvious, we concluding by showing how this actually fails in some larger sets of numbers, such as the rational numbers and a set of algebraic integers, where the number 9 can be factored two ways: 9=3*3=(2+sqrt{-5})(2-sqrt{-5}).

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