# Metamath Proof Explorer

## Table of Contents - 6. ELEMENTARY NUMBER THEORY

Here we introduce elementary number theory, in particular the elementary
properties of divisibility and elementary prime number theory.

- Elementary properties of divisibility
- Irrationality of square root of 2
- Some Number sets are chains of proper subsets
- The divides relation
- Even and odd numbers
- The division algorithm
- Bit sequences
- The greatest common divisor operator
- Bézout's identity
- Algorithms
- Euclid's Algorithm
- The least common multiple
- Coprimality and Euclid's lemma
- Cancellability of congruences

- Elementary prime number theory
- Elementary properties
- Coprimality and Euclid's lemma (cont.)
- Properties of the canonical representation of a rational
- Euler's theorem
- Arithmetic modulo a prime number
- Pythagorean Triples
- The prime count function
- Pocklington's theorem
- Infinite primes theorem
- Sum of prime reciprocals
- Fundamental theorem of arithmetic
- Lagrange's four-square theorem
- Van der Waerden's theorem
- Ramsey's theorem
- Primorial function
- Prime gaps
- Decimal arithmetic (cont.)
- Cyclical shifts of words (cont.)
- Specific prime numbers
- Very large primes