Elementary Number Theory

Forward We start with the set of natural numbers, N = {1, 2, 3, . . .} equipped with the familiar addition and multiplication and assume that it satisfies the induction axiom. It allows us to establish division with a residue and the Euclid’s algorithm that computes the greatest commond divisor of two natural numbers. It also leads to a proof of the fundamental theorem of arithmetic: Every natural number is a product of prime numbers in a unique way up to the order of the factors. Euclid’s theorem about the infinitude of the prime numbers is a consequence of that theorem. Next we introduce congruences and the Euler’s φ-function (φ(n) is the number of the natural numbers between 1 and n that are relatively prime to n). Then we prove Euler’s theorem: a ≡ 1 mod n for each natural number n and every integer a relatively prime to n. We also prove the Chinese remainder theorem and conclude the multiplicity of the Euler phi function: φ(mn) = φ(m)φ(n) if m and n are relatively prime. This theorem is the main ingredient in the first and most applied public key in cryptography. Next we prove that each prime number p has φ(p− 1) primitive roots modulo prime numbers. Most of this material enters into the proof of the quadratic reciprocity law: Every distinct odd prime numbers p, q satisfy ( q p ) = (−1) p−1 2 q−1 2 (p q ) , where ( q p ) = 1 if there exists an integer x with x ≡ q mod p, and −1 otherwise. The last part of these notes is devoted to the proof of Dirichlet’s theorem about the Dirichlet density of the set prime numbers p ≡ a mod m, where gcd(a, m) = 1.