# prologue

Number theory is the study of integers. The integers, denoted by $\mathbb{Z}$ are the counting numbers $1, 2, 3, \dots$, their negatives $-1, -2, -3, \dots$, and zero: $$\mathbb{Z} = \{ \dots -3, -2, -1, 0, 1, 2, 3, \dots \}.$$

We will assume the familiar arithmetic of addition, subtraction, and multiplication of integers; integer division will be the first topic in this treatment.

We will also assume the principle of mathematical induction and the well-ordering principle.

The integers and their arithmetic are based on the Peano axioms , which uniquely define the natural numbers $$\mathbb{N} = \{ 0, 1, 2, 3, \dots \}.$$ The integers $\mathbb{Z}$ are then defined by extending the natural numbers so that each natural number $n$ has an additive inverse $-n$ satisfying $$n + (-n) = 0.$$

Some treatments of number theory will define the natural numbers as beginning at $1$. The original Peano axioms did just that. There is no consensus in the mathematical community, but the Peano axioms section of the appendix defines them as starting at $0$. To avoid confusion throughout the text, we will denote by $\mathbb{N}_k$ the natural numbers starting at the integer $k$, so \begin{align} \mathbb{N}_0 &= \{ 0, 1, 2, 3, \dots \} \\ \mathbb{N}_1 &= \{ 1, 2, 3, 4, \dots \} \\ \mathbb{N}_5 &= \{ 5, 6, 7, 8, \dots \} \\ \vdots \end{align}

The sections in the appendix linked above are introductory and not comprehensive. The interested reader might augment those topics with additional sources.