wiki/math/number_theory/notes.page

---
format: markdown
toc: yes
title: Number Theory Study Notes
...

# Texts

- *An Introduction to the Theory of Numbers*, Hardy and Wright

# Reading Notes

## *Hardy and Wright*, 24 Jan 2014

### Notation

$\to$
: implies

$\equiv$
: is equivalent to

$\exists$
: there is an

$\in$
: relation of a member of a class to a class

A * affixed to a theorem number means the proof was too difficult to include in the book.

### The Series of Primes

$..., -3, -2, -1, 0, 1, 2, ...$
: The *rational integers* or just *integers*

$0, 1, 2, 3, ...$
: The *non-negative integers*

$1,2,3,...$
: The *positive integers* -- these are the primary subject of number theory (a.k.a "arithmetic")

An integer $a$ is *divisible* by a non-zero integer $b$ if there is a $c$ such that

$$a = bc$$

If $a$ and $b$ are positive, so is $c$

$b\mid a$
: $b$ is a *divisor* of $a$ ($a$ is divisible by $b$)

$b\nmid a$
: $b$ is not a divisor of $a$

For all $a$, $1\mid a$ and $a\mid a$; For non-zero $b$, $b\mid 0$.

When $c\neq0$:
$$\begin{align}
b\mid a \cdot c\mid b &\to c\mid a\\
b\mid a &\to bc\mid ac\\
\end{align}$$
and
$$c\mid a \cdot c\mid b \to c\mid ma + nb$$
for all integers $m$ and $n$

A number $p$ is *prime* if:

i. $p > 1$
ii. $p$ has no positive divisors except $1$ and $p$

The number $1$ is not considered prime.

A number is *composite* if it is greater than $1$ and not prime.

**Theorem 1**
: Every positive integer, except $1$, is a product of primes.