Number Theory, Lecture 1 Number Theory, Lecture 1 Jan Snellman Integers, Divisibility, Primes Divisibility Definition Elementary properties Partial order 1 Prime number Jan Snellman Division Algorithm Greatest 1Matematiska Institutionen common divisor Link¨opingsUniversitet Definition Bezout Euclidean algorithm Extended Euclidean Algorithm Unique factorization into primes Some Lemmas An importan property of primes Euclid, again Fundamental theorem of arithmetic Exponent vectors Least common multiple More about primes Sieve of Eratosthenes Primes in arithmetic progressions Link¨oping,spring 2019 Lecture notes availabe at course homepage http://courses.mai.liu.se/GU/TATA54/ Number Summary Theory, Lecture 1 Jan Snellman Divisibility Definition Elementary Extended Euclidean Algorithm properties 1 Divisibility Partial order 3 Unique factorization into primes Prime number Definition Division Algorithm Some Lemmas Elementary properties Greatest An importan property of primes common Partial order divisor Euclid, again Definition Prime number Bezout Fundamental theorem of arithmetic Euclidean algorithm Division Algorithm Extended Euclidean Exponent vectors Algorithm Unique 2 Greatest common divisor Least common multiple factorization into primes Definition 4 More about primes Some Lemmas An importan Bezout Sieve of Eratosthenes property of primes Euclid, again Euclidean algorithm Primes in arithmetic progressions Fundamental theorem of arithmetic Exponent vectors Least common multiple More about primes Sieve of Eratosthenes Primes in arithmetic progressions Number Summary Theory, Lecture 1 Jan Snellman Divisibility Definition Elementary Extended Euclidean Algorithm properties 1 Divisibility Partial order 3 Unique factorization into primes Prime number Definition Division Algorithm Some Lemmas Elementary properties Greatest An importan property of primes common Partial order divisor Euclid, again Definition Prime number Bezout Fundamental theorem of arithmetic Euclidean algorithm Division Algorithm Extended Euclidean Exponent vectors Algorithm Unique 2 Greatest common divisor Least common multiple factorization into primes Definition 4 More about primes Some Lemmas An importan Bezout Sieve of Eratosthenes property of primes Euclid, again Euclidean algorithm Primes in arithmetic progressions Fundamental theorem of arithmetic Exponent vectors Least common multiple More about primes Sieve of Eratosthenes Primes in arithmetic progressions Number Summary Theory, Lecture 1 Jan Snellman Divisibility Definition Elementary Extended Euclidean Algorithm properties 1 Divisibility Partial order 3 Unique factorization into primes Prime number Definition Division Algorithm Some Lemmas Elementary properties Greatest An importan property of primes common Partial order divisor Euclid, again Definition Prime number Bezout Fundamental theorem of arithmetic Euclidean algorithm Division Algorithm Extended Euclidean Exponent vectors Algorithm Unique 2 Greatest common divisor Least common multiple factorization into primes Definition 4 More about primes Some Lemmas An importan Bezout Sieve of Eratosthenes property of primes Euclid, again Euclidean algorithm Primes in arithmetic progressions Fundamental theorem of arithmetic Exponent vectors Least common multiple More about primes Sieve of Eratosthenes Primes in arithmetic progressions Number Summary Theory, Lecture 1 Jan Snellman Divisibility Definition Elementary Extended Euclidean Algorithm properties 1 Divisibility Partial order 3 Unique factorization into primes Prime number Definition Division Algorithm Some Lemmas Elementary properties Greatest An importan property of primes common Partial order divisor Euclid, again Definition Prime number Bezout Fundamental theorem of arithmetic Euclidean algorithm Division Algorithm Extended Euclidean Exponent vectors Algorithm Unique 2 Greatest common divisor Least common multiple factorization into primes Definition 4 More about primes Some Lemmas An importan Bezout Sieve of Eratosthenes property of primes Euclid, again Euclidean algorithm Primes in arithmetic progressions Fundamental theorem of arithmetic Exponent vectors Least common multiple More about primes Sieve of Eratosthenes Primes in arithmetic progressions Number Integers, divisibility Theory, Lecture 1 Jan Snellman Divisibility Definition Definition Elementary properties • Z = f0; 1; -1; 2; -2; 3; -3;:::g Partial order Prime number • = f0; 1; 2; 3;:::g Division Algorithm N Greatest • = f1; 2; 3;:::g common P divisor Definition Bezout Unless otherwise stated, a; b; c; x; y; r; s 2 , n; m 2 . Euclidean algorithm Z P Extended Euclidean Algorithm Definition Unique factorization ajb if exists c s.t. b = ac. into primes Some Lemmas An importan property of primes Example Euclid, again Fundamental theorem of arithmetic 3j12 since 12 = 3 ∗ 4. Exponent vectors Least common multiple More about primes Sieve of Eratosthenes Primes in arithmetic progressions Number Theory, Lecture 1 Jan Snellman Lemma Divisibility Definition • Elementary aj0, properties Partial order • 0ja a = 0, Prime number Division Algorithm • 1ja, Greatest common • () divisor aj1 a = ±1, Definition Bezout • ajb ^ bja a = ±b Euclidean algorithm () Extended Euclidean • Algorithm ajb -ajb aj-b Unique • ajb ^ ajc =() aj(b + c), factorization into primes () () Some Lemmas • ajb = ajbc. An importan property of primes ) Euclid, again Fundamental theorem of ) arithmetic Exponent vectors Least common multiple More about primes Sieve of Eratosthenes Primes in arithmetic progressions Number Theory, Theorem Lecture 1 Retricted to P, divisibility is a partial order, with unique minimal element 1. Jan Snellman Divisibility Definition Part of Hasse diagram Id est, Elementary properties 1 aja, Partial order Prime number 2 ajb ^ bjc = ajc, Division Algorithm Greatest 3 ajb ^ bja = a = b. common 4 6 9 10 15 14 21 35 divisor ) Definition ) Bezout Euclidean algorithm Extended Euclidean Algorithm 2 3 5 7 Unique factorization into primes Some Lemmas An importan property of primes Euclid, again 1 Fundamental theorem of arithmetic Exponent vectors Least common multiple More about primes Sieve of Eratosthenes Primes in arithmetic progressions Number Theory, Lecture 1 Jan Snellman Divisibility Definition Definition Elementary properties Partial order n 2 P is a prime number if Prime number Division Algorithm • n > 1, Greatest common • mjn = m 2 f1; ng divisor Definition (positive divisors, of course -1; -n also divisors) Bezout Euclidean algorithm ) Extended Euclidean Algorithm Unique 2; 3; 5; 7; 11; 13; 17; 19; 23; 29; 31;::: factorization into primes Some Lemmas An importan property of primes Euclid, again Fundamental theorem of arithmetic Exponent vectors Least common multiple More about primes Sieve of Eratosthenes Primes in arithmetic progressions Number Division algorithm Theory, Lecture 1 Jan Snellman Divisibility Definition Elementary properties Partial order Theorem Prime number Division Algorithm a; b 2 Z, b 6= 0. Then exists unique k; r, quotient and remainder, such that Greatest common • a = kb + r, divisor Definition • 0 ≤ r < b. Bezout Euclidean algorithm Extended Euclidean Algorithm Example Unique factorization into primes -27 = (-6) ∗ 5 + 3. Some Lemmas An importan property of primes Euclid, again Fundamental theorem of arithmetic Exponent vectors Least common multiple More about primes Sieve of Eratosthenes Primes in arithmetic progressions Number Proof, existence Theory, Lecture 1 Jan Snellman Divisibility Suppose a; b > 0. Fix b, induction over a, base case a < b, then Definition Elementary properties Partial order a = 0 ∗ b + a: Prime number Division Algorithm Greatest Otherwise common divisor a = (a - b) + b Definition Bezout Euclidean algorithm and ind. hyp. gives Extended Euclidean 0 0 0 Algorithm a - b = k b + r ; 0 ≤ r < b Unique factorization into primes so Some Lemmas 0 0 0 0 An importan a = b + k b + r = (1 + k )b + r : property of primes Euclid, again 0 0 Fundamental Take k = 1 + k , r = r . theorem of arithmetic Exponent vectors Least common multiple More about primes Sieve of Eratosthenes Primes in arithmetic progressions Number Proof, uniqueness Theory, Lecture 1 Jan Snellman Divisibility Definition Elementary properties If Partial order Prime number a = k b + r = k b + r ; 0 ≤ r ; r < b Division Algorithm 1 1 2 2 1 2 Greatest common then divisor Definition 0 = a - a = (k1 - k2)b + r1 - r2 Bezout Euclidean algorithm Extended Euclidean hence Algorithm (k1 - k2)b = r2 - r1 Unique factorization into primes jRHSj < b, so jLHSj < b, hence k1 = k2. But then r1 = r2. Some Lemmas An importan property of primes Euclid, again Fundamental theorem of arithmetic Exponent vectors Least common multiple More about primes Sieve of Eratosthenes Primes in arithmetic progressions Number Theory, Lecture 1 Jan Snellman Divisibility Example Definition Elementary properties a = 23,b = 5. Partial order Prime number Division Algorithm 23 = 5 + (23 - 5) = 5 + 18 Greatest common = 5 + 5 + (18 - 5) = 2 ∗ 5 + 13 divisor Definition Bezout = 2 ∗ 5 + 5 + (13 - 5) = 3 ∗ 5 + 8 Euclidean algorithm Extended Euclidean = 3 ∗ 5 + 5 + (8 - 5) = 4 ∗ 5 + 3 Algorithm Unique factorization k = 4, r = 3. into primes Some Lemmas An importan property of primes Euclid, again Fundamental theorem of arithmetic Exponent vectors Least common multiple More about primes Sieve of Eratosthenes Primes in arithmetic progressions Number Greatest common divisor Theory, Lecture 1 Jan Snellman Divisibility Definition Elementary properties Partial order Definition Prime number Division Algorithm a; b 2 . The greatest common divisor of