64 lines
1.2 KiB
Plaintext
64 lines
1.2 KiB
Plaintext
---
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format: markdown
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toc: yes
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title: Number Theory Study Notes
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...
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# Texts
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- *An Introduction to the Theory of Numbers*, Hardy and Wright
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# Reading Notes
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## *Hardy and Wright*, 24 Jan 2014
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### Notation
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$\to$
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: implies
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$\equiv$
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: is equivalent to
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$\exists$
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: there is an
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$\in$
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: relation of a member of a class to a class
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A * affixed to a theorem number means the proof was too difficult to include in the book.
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### The Series of Primes
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$..., -3, -2, -1, 0, 1, 2, ...$
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: The *rational integers* or just *integers*
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$0, 1, 2, 3, ...$
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: The *non-negative integers*
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$1,2,3,...$
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: The *positive integers* -- these are the primary subject of number theory (a.k.a "arithmetic")
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An integer $a$ is *divisible* by a non-zero integer $b$ if there is a $c$ such that
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$$a = bc$$
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If $a$ and $b$ are positive, so is $c$
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$b\mid a$
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: $b$ is a *divisor* of $a$ ($a$ is divisible by $b$)
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$b\nmid a$
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: $b$ is not a divisor of $a$
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For all $a$, $1\mid a$ and $a\mid a$; For non-zero $b$, $b\mid 0$.
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When $c\neq0$:
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$$\begin{align}
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b\mid a \cdot c\mid b &\to c\mid a\\
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b\mid a &\to bc\mid ac\\
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\end{align}$$
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and
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$$c\mid a \cdot c\mid b \to c\mid ma + nb$$
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for all integers $m$ and $n$
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