Number Theory
These are notes on elementary number theory; that is, the part of number theory which does not involves methods from abstract algebra or complex variables.
The first link in each item is to a Web page; the second is to a PDF file. Use the PDF if you want to print it. There are videos for some of the sections, but they don't cover everything in the printed notes.
[April 30, 2019] These notes (but not necessarily the videos) were revised in Spring, 2019. I'm planning to add some material, may over the summer (for instance, more on periodic continued fractions and the FermatPell equations).
 The ring of integers.
[PDF]
 The greatest integer function.
[PDF]
 Sums and products.
[PDF]
Sums (video: 17 minutes, 20 MB)
Products (video: 20 minutes, 23 MB)
 Binomial coefficients.
[PDF]
 Induction.
[PDF]
Induction  Part 1 (video: 28 minutes, 27 MB)
Induction  Part 2 (video: 11 minutes, 13 MB)
Induction  Part 3 (video: 11 minutes, 14 MB)
 Fibonacci numbers.
[PDF]
Fibonacci numbers  Part 1 (video: 11 min., 13 MB)
Fibonacci numbers  Part 2 (video: 22 min., 27 MB)
 Divisibility and the Division Algorithm.
[PDF]
Divisibility and the Division Algorithm (video: 37 minutes, 44 MB)
 Prime numbers.
[PDF]
Prime numbers (video: 33 minutes, 38 MB)
 Greatest common divisors.
[PDF]
 The Extended Euclidean Algorithm.
[PDF]

The Fundamental Theorem of Arithmetic.
[PDF]
 Elementary factoring methods; Fermat factorization.
[PDF]
Fermat factorization (video: 15 minutes, 18 MB)
 Fermat numbers.
[PDF]

Linear Diophantine equations.
[PDF]

Modular arithmetic.
[PDF]
Modular arithmetic  Part 1 (video: 25 min., 28 MB)
Modular arithmetic  Part 2 (video: 30 min., 34 MB)
Modular arithmetic  Part 3 (video: 19 min., 21 MB)
 The dayoftheweek algorithm.
[PDF]

Nonlinear Diophantine equations  some examples
[PDF]

Solving linear congruences (one and two variables).
[PDF]

The Chinese Remainder Theorem.
[PDF]

Systems of linear congruences.
[PDF]

Prime power congruences.
[PDF]
 Wilson's Theorem and
Fermat's Theorem.
[PDF]
 Euler's Theorem; the Euler phi function.
[PDF]
Euler's theorem; the Euler phifunction (video: 23 minutes, 27 MB)

Arithmetic functions; multiplicative functions; Dirichlet products; Möbius inversion; properties of the Euler phi function.
[PDF]

The sum and number of divisors functions.
[PDF]
 Perfect numbers and Mersenne primes.
[PDF]

Character and block ciphers.
[PDF]

Exponential ciphers; the RSA algorithm.
[PDF]
 Quadratic residues.
[PDF]
 Quadratic reciprocity.
[PDF]

The Jacobi symbol.
[PDF]
 Fractions and rational numbers; bases
[PDF]
 Finite continued fractions.
[PDF]

Infinite continued fractions.
[PDF]

Approximation by rationals.
[PDF]

Periodic continued fractions (only half of Lagrange's theorem right now).
[PDF]

The FermatPell equation (examples, no proofs yet).
[PDF]
Contact information
Bruce Ikenaga's Home Page