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.

*[December 14, 2022]* I updated the Chinese Remainder Theorem notes with a proof of the last theorem (on non-relatively prime moduli).

*[August 11, 2022]* I clarified the assumptions in many of the results on finite continued fractions (so all the a's are positive reals except that a_{0} can be nonnegative), and added a part to the last example.

*[June 28, 2019]* These notes were revised in Spring, 2019. I revised the sections on infinite continued fractions and periodic continued fractions after the term during May and June. I'm planning to add sections on purely periodic continued fractions and continued fractions for radicals, and expand the sections on the Fermat-Pell equation.

- The ring of integers. [PDF]
- The greatest integer function. [PDF]
- Sums and products. [PDF]
- Binomial coefficients. [PDF]
- Induction. [PDF]
- Fibonacci numbers. [PDF]
- Divisibility and the Division Algorithm. [PDF]
- Prime numbers. [PDF]
- Greatest common divisors. [PDF]
- The Extended Euclidean Algorithm. [PDF]
- The Fundamental Theorem of Arithmetic. [PDF]
- Elementary factoring methods; Fermat factorization. [PDF]
- Fermat numbers. [PDF]
- Linear Diophantine equations. [PDF]
- Modular arithmetic. [PDF]
- The day-of-the-week 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]
- 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; Lagrange's theorem [PDF]
- Purely periodic continued fractions; Galois's theorem [PDF]
- The Fermat-Pell equation (examples, no proofs yet). [PDF]