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title: Number Theory Study Notes
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# 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.