0.2 A Taste of Number Theory

Objectives:

•

Introduce the notion of proof through basic number theory
•

Recognize the struggles of proving theorems rigorously

0.2.1 Even and Odd Integers

Before we get into the basics of propositional logic, let’s start by trying to write a few proofs ourselves. But what should we write proofs about?

It is typical in a first proof-writing course to start with number theory. The discoveries we will make here date back to ancient civilizations - it seems like the right place to begin.

We will fix some of our notations to start. We write the set of integers as

\mathbb{Z}:=\{\dots,-3,-2,-1,0,1,2,3,\dots\},

and we write the set of natural numbers as

\mathbb{N}:=\{1,2,3,\dots\}.

Definition 0.2.1.

An integer $n$ is even if $n=2k$ for some $k\in\mathbb{Z}$ . An integer $n$ is odd if $n=2k+1$ for some $k\in\mathbb{Z}$ .

This definition introduces a new symbol: we will write $\in$ to denote “is an element of” or “is a member of”. So in English, our first sentence reads, “An integer $n$ is even if $n$ is twice some number $k$ , where $k$ is an element of the integers.”

Theorem 0.2.2.

If $n$ is an even integer, then $n^{2}$ is an even integer.

Theorem 0.2.3.

The sum of two consecutive integers is odd.

Notice that to prove these two theorems we had to start general. In Section 0-1 we saw that “proofs by example” don’t hold up to the standard of mathematical rigor. We need to make an argument that holds for all examples we could consider. Hence for our previous theorem, our “two consecutive integers” weren’t just two numbers we chose like “6” and “7”; they were $n$ and $n+1$ for some integer $n$ . We could have also chosen $n-1$ and $n$ as our two consecutive integers, or even $n+2$ and $n+3$ , as these descriptions fit the general description of any two consecutive integers.

However, to disprove a general statement, like “All cars are red,” we only need to show a counterexample. The existence of one blue car is enough to say that the statement is false. It is easy to fall into a trap here, as you might be tempted to say “there is no reason all cars have to be red”. But this doesn’t disprove the statement. Being suspicious about a whether a statement is true doesn’t mean that the statement is false; it must be disproven with a counterexample.

Example 0.2.4.

Determine whether the statement below is true or false. If the statement is true, prove it. If the statement is false, provide a counterexample.

The product of an odd integer and an even integer is odd.

0.2.2 Division of Integers

Number theory finds its roots in the study of prime numbers. Euclid was fascinated by these and proved that there were infinitely many prime numbers. We will discuss this in Section 6.2. To discuss prime numbers, we have to discuss what it means for one number to divide evenly into another:

Definition 0.2.5.

Given $n,m\in\mathbb{Z}$ , we say that $n$ divides $m$ , written $n|m$ , if there exists $k\in\mathbb{Z}$ such that $m=nk$ . If $n|m$ , we may also say that $m$ is divisible by $n$ or that $n$ is a factor of $m$ .

Theorem 0.2.6.

The sum of any three consecutive integers is always divisible by three.

Example 0.2.7.

Determine whether the statements below are true or false. If a statement is true, prove it. If a statement is false, provide a counterexample.

If $a,n\in\mathbb{Z}$ such that $a$ divides $n$ , then $a$ divides $n^{2}$ .

If $a,n\in\mathbb{Z}$ such that $a$ divides $n^{2}$ , then $a$ divides $n$ .

The mini-statements “ $a$ divides $n$ ” and “ $a$ divides $n^{2}$ ” are called predicates in logic. Let’s call “ $a$ divides $n$ ” Predicate 1 and “ $a$ divides $n^{2}$ ” Predicate 2. Then both statemenets written above are written in the form “If [Statement x], then [Statement y].” These are called conditional propositions. What’s more, the second statement is the converse of the first, as it flips the order of the two predicates.

We see from the above example that it is not necessarily the case that flipping the order of predicates keeps the statement true. Let’s investigate a bit more:

Example 0.2.8.

Is the converse of Theorem 0.2.6 true? Prove or disprove.

Notice that this is the converse of Theorem 0.2.6. Even though Theorem 0.2.6 is not written using the exact “If… then…” English terms, it can be rewritten into that form using a rephrasing: “If you have the sum of three consecutive integers, then that integer is divisible by three.”

For this next problem, we will need to use the following fact: “Every integer is either odd or even," which we will assume without proof for now. (The proof uses complete induction, which we will cover in Section 3.2.) In the problem below: since $n=2k+1$ for some integer $k$ by definition of odd, see if you can apply the fact above to the integer $k$ .

Theorem 0.2.9.

If $n\in\mathbb{Z}$ such that $n$ is odd, then 8 divides $n^{2}-1$ .