Read chap 1 The natural numbers [p. 1 - 20]

Notes for chap 1.1 Calculation with integers [p. 1 - 9]

Notes for chap 1.2 The infinitude of the number system [p. 9 - 20]

Review

The topic of congruences, first explored by Gauss, has two notable features. One is the geometric representation as integral points on a circle, with the number of points corresponding to the modulus (i.e. base). The second is some interesting theorems when one considers a prime number for the modulus.

Computation in any base is fairly straight forward if one first writes out the appropriate addition and multiplication tables to make sure that one is using the symbols properly.

The Fundamental Theorem of Arithmetic which states that every positive integer can be factored into a unique product of prime integers is both a "neat" theorem as well as a very powerful one, leading to many additional properties.

Another important theorem is known as The Prime Number Theorem. It was originally a conjecture by Gauss (1793?) and was finally proven in 1896 by two French mathematicians although Riemann (1859) is credited with mapping out the basic strategy of the proof. The Prime Number Theorem states that the distribution of prime numbers (i.e. the ratio p(n)/n ) is approximately 1/ln(n).

Content

Courant & Robbins Supplement to Chapter 1 "The Theory of Numbers" [p. 21 - 51]

Fermat is primarily known for two theorems in number theory. One is called Fermat's Theorem and the other is referred to as Fermat's Last Theorem.

Fermat's Theorem (aka Fermat's Little Theorem).

If p is any prime which does not divide the integer a, then . This means that the (p-1)st power of a leaves a remainder 1 upon division by p.

It is not obvious to me why this is considered important. I think it has something to do with testing for whether a number is prime or not. The next topic of quadratic residues is related to this as well. This appears to be a major topic in number theory.

While googling quadratic residues I noticed that the topic was related to cryptography and that brought to mind the book In Code (2001) by Sarah Flannery which describes her personal journey into crytography. A glance at the index to this book failed to show quadratic residues but it did mention Fermat's Little Theorem and a glance at the contents showed that his book contains a superb introduction to the topic. I think I will give this a closer look before returning to Courant & Robbins.

Fermat's Last Theorem

has no non-zero integer solutions for x, y, and z when n>2.

This was finally proven in 1995 by an English mathematician Andrew Wiles.

Prime Numbers

Algebraic relationships

Distribution of prime numbers

The Prime Number Theorem

Congruences

Geometric representation

Rules for divisibility

Fermat's Theorem

Quadratic residues

Pythagorean Numbers

Fermat's Last Theorem

Division

Euclidean algorithm

Diophantine equations

Euler's phi function