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Chapter 1 Prologue

What is number theory? Briefly, it is the study of the integers and questions arising from them.

Definition 1.0.1. The Integers.

The set of counting numbers is denoted

\begin{equation*} \mathbb{N} = \{0,1,2,3,4,\cdots\}\text{.} \end{equation*}

Note that in this text, this set begins at zero 1 . The integers is the set of positive and negative counting numbers:

\begin{equation*} \mathbb{Z} = \{ \cdots ,-3,-2,-1,0,1,2,3,\cdots \}\text{.} \end{equation*}
You can search Mathematics Stack Exchange, Wikipedia, and many other interesting sites for discussions about this. Authors disagree, though number theory texts tend to go with the older tradition of only counting positive integers among the “natural numbers”, both because they count things and because they are a natural set to work with. With the advent of computers and (often) zero-based counting, as well as set theory, there is more variety, and it will be convenient to start at zero here since we integrate the use of a zero-based computer language so much. Apparently the ISO standard also begins counting at zero.

This is a fairly dry definition, though. The best way to find out what this definition means is to try to answer some questions about integers!