New Zealand Maths Olympiad Committee online

The fact that, for every positive integer n, there is a prime between n and 2n is known as Bertrand’s postulate (which is a bit odd, as it’s a theorem, but anyhow …) It arises occasionally in Olympiad style problems (usually with the note “You may assume Bertrand’s Postulate that …”) Michael Nielsen has a nice post giving an elementary proof at the Polymath wiki.

Michael

(At least!) a couple of good, comprehensive introductions to elementary number theory are available online.  These notes by Jim Hefferon and W. Edwin Clark are nicely written and gently-paced.  These ones by Naoki Sato are a bit more Olympiad-focused.

This series of short introductory articles by Arkadii Slinko covers some of the most fundamental results in number theory.

Tutorial 1: Divisibility and Primes

Tutorial 2: The Euclidean Algorithm

Tutorial 3: Euler’s Function

Tutorial 4: Primes that are Sums of Two Squares

Tutorial 5: Bertrand’s Theorem

(Update, 24/1/09:  some typos fixed.)

These notes by Arkadii Slinko outline a number of techniques for solving Diophantine equations.

Solutions for some of the problems are available, and can be obtained by writing to nzmathsolymp@gmail.com.

(Update, 24/1/2009:  some typos fixed.)