More notes

math/number_theory/notes.page

@@ -14,7 +14,7 @@ title: Number Theory Study Notes
 
 ### Notation
 
-$\rightarrow$
+$\to$
 : implies
 
 $\equiv$
@@ -26,4 +26,38 @@ $\exists$
 $\in$
 : 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.
+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 $c\neq0$:
+$$\begin{align}
+b\mid a . c\mid b &\to c\mid a\\
+b\mid a &\to bc\mid ac\\
+\end{align}$$
+and
+$$c\mid a . c\mid b \to c\mid ma + nb$$
+for all integers $m$ and $n$