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.
[July 15, 2019] These notes (but not necessarily the videos) were revised in Spring, 2019. I revised the sections on infinite continued fractions and periodic continued fractions after the term during May and June, and added a section on purely periodic continued fractions in July. I'm planning to add a section on continued fractions for radicals, and expand the section on the FermatPell equation.
 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; the algorithm for the continued fraction of a
quadratic irrational
[PDF]

Purely periodic continued fractions; Galois's theorem
[PDF]

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