I'm very excited to attempt venturing into the world of mathematics on my own. I have taken a few math courses at my university, but I honestly need some work. It seems like I'm much slower at grasping the concepts of the courses than my fellow peers, so I've decided to allocate some spare time to studying Elementary Number Theory by Gareth A. Jones and J. Mary Jones for the extra practice. I hope you find my notes useful! I aim to be as thorough as possible in my notes in order to demonstrate to myself that I fully understand the subject area.

I don't indent on using LaTeX because it seems awfully boring to learn. I will use a combination of typed notes and hand-written notes (that are hopefully somewhat legible).

I plan on skipping exercises that look boring or tedious. But enough talk, it's time to do some math! The first topic of the book is divisibility.

1.1 Divisors

We dive right in to talking about the division algorithm.

Theorem 1.1

If a and b are integers with b > 0, then there exist a unique pair of integers q and r such that a = bq+r and 0 <= r < b.

Example 1.1

Note that r is always greater than or equal to 0 and strictly less than b.

Proving the division algorithm (Theorem 1.1)

The proof of the division algorithm takes advantage of something called the well-ordering principle.

The well-ordering principle

The well-ordering_principle simply states that every non-empty set of positive integers contains a least element. So take any arbitrary set of positive integers:

• {5,3,9,12}
• {2,3}
• {1,1,1}

For each and every set of positive integers you could ever come up with, there will always be some number in the set that's the smallest.

Back to the proof

Corollary 1.2

If a and b are integers with b not equal to 0, then there exists unique q and r such that a = qb + r with 0 <= r < |b|.

Example 1.2

To demonstrate usage of the division algorithm, we need to prove that if n is a square, then n leaves a remainder of 0 or 1 when divided by 4.

What makes the division algorithm so useful is that, as demonstrated in this proof, we can easily take a number and put it in terms of any other number. We were able to directly associate a with the number 4 thanks to the division algorithm guaranteeing a unique q and r for any provided a and b>0.

Exercise 1.1

Find a shorter proof for Example 1.2, based on using b=2 in the division algorithm

Here's my attempt:

Definition of "divides"

If a and b are any integers, and a = qb for some integer q, then we say that b divides a, or b is a factor of a, or a is a multiple of b. When b divides a, we write b|a. Every integer divides 0 (0 = 0b for all b), and 1 divides every integer.

Exercise 1.3

Prove the following

a) if a|b and b|c, then a|c

b) if a|b and c|d, then ac|bd

c) if m is not 0, then a|b iff ma|mb

d) if d|a and a is not 0, then |d| <= |a|