Chapter 1 Prologue
Definition 1.0.1. The Integers.
The set of counting numbers is denoted
Note that in this text, this set begins at zero 1 . The integers is the set of positive and negative counting numbers:
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.
Summary: Prologue
After reminding ourselves of The Integers, this introductory chapter covers the following main topics.
In Question 1.1.1 we introduce the notion of the conductor to get thinking about nontrivial integer questions.
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We review basic uses of the following principles:
Proofs by Induction
Basic facts about Divisibility, of which we will especially use Proposition 1.2.8
We get a brief look at where we are going in this text.
Finally, after the usual Exercises, there are few notes on Using Sage for Interactive Computation.