Introduction

Maggie Schildt

29 Introduction

We have hinted at some of the aspects of number theory already. In the previous section we kept bringing up this idea of being able to simplify terms using multiples of a base value in say $\mathbb{Z}$ , this falls under congruence which will discuss in this section. We have also seen number theory in every aspect of each chapter of mathematics thus far. When we were introduced to field axioms we were looking at what number theory calls the laws of arithmetic.

Number Theory is considered to be a higher realm of arithmetic, just as Modern Algebra might be considered a higher realm of algebra. In number theory we study the structures and properties of integers. We say any element $a$ in the set $\mathbb{Z}$ is an integer. An integer is an number, positive or negative, that is a whole number. We are very familiar with this set as we used it tirelessly in our examples for Abstract Algebra. In this section we will cover the different relationships and structures of integers [39].

Theorem: Lagrange Theorem

Every natural number can be represented as the sum of four integers squares.

This theorem can be represented by $p=a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2}$ . It might be better to consider this a higher order axiom of integers. The same goes for Pythagorean Theorem which is given as follows [39].

Theorem: Pythagorean Theorem

For a right triangle with legs $a$ and $b$ and a hypotenuse $c$ it follows that $a^{2}+b^{2}=c^{2}$ such that $c=\sqrt{a^{2}+b^{2}}$ .

We should already be very familiar with this theorem as it should have been introduced to us in geometry but since it is a theorem pertaining to integers we include here as well. Having a general idea of the topic of number theory we progress to some of the main topics of higher arithmetic, beginning with a discussion on primes.

29 Introduction

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