Learning: The Journey of a Lifetime
or A Cloud Chamber of the Mind

September 2005 Mathematics Learning Log

An Example of a "Learning Process" Journal

Tuesday October 04, 2005
Learning Log Number 20

5:40 am Lethbridge, Alberta

An early start to the day.

What is Mathematics? (1941) Richard Courant & Herbert Robbins      

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

Sept 26 Sept 29 Yes

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

Sept 26 Sept 26 Yes

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

Sept 26 Sept 26 Yes
Read chap 1 supplement The theory of numbers [p. 21 - 51] Sept 29    
Notes for chap 1 supplement 1.1 The prime numbers [p. 21 - 31] Sept 29 Oct 01 Yes
Notes for chap 1 supplement 1.2 Congruences [p. 31 - 40] Oct 03    
Notes for chap 1 supplement 1.3 Pythagorean numbers and Fermat's Last Theorem [p. 40 - 42] Oct 04    
Notes for chap 1 supplement 1.4 The Euclidean Algorithm [p. 42 - 51]      
Calculus (1994) Michael Spivak      
Read chap 1 Basic properties of numbers [p. 3 - 20]      
Notes for chap 1      
Exercises for chap 1 [p. 13 - 20]      
Prime Obsession (2003) John Derbyshire      
Reread chap 1 - 8 [p. 1 - 136]      


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).

Yesterday I made some hand written notes on congruences and completed a couple of exercises.


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.

Number Theory Sept 30 Sept 30
Prime Numbers
Sept 30 Sept 30
Algebraic relationships
Sept 30 Sept 30

Distribution of prime numbers

Sept 30 Sept 30

The Prime Number Theorem

Sept 30 Sept 30
Sept 30 Sept 30

Geometric representation

Sept 30 Sept 30

Rules for divisibility

Sept 30 Oct 03

Fermat's Theorem

Oct 03 Oct 04

Quadratic residues

Oct 04  
Pythagorean Numbers

Fermat's Last Theorem


Euclidean algorithm


Diophantine equations


Euler's phi function


My sudden interest in the book "In Code" by Sarah Flannery is typical of the way in which I continue to zig and zag. As soon as I hit a difficulty I look for other sources (both online and on my bookshelves).

7:10 am

Total elapsed time for the day: 1 hr. 30 min