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 = {0, 1, −1, 2, −2, 3, −3,...} Partial order Prime number • = {0, 1, 2, 3,...} Division Algorithm N Greatest • = {1, 2, 3,...} common P divisor Definition Bezout Unless otherwise stated, a, b, c, x, y, r, s ∈ , n, m ∈ . Euclidean algorithm Z P Extended Euclidean Algorithm Definition Unique factorization a|b if exists c s.t. b = ac. into primes Some Lemmas An importan property of primes Example Euclid, again Fundamental theorem of arithmetic 3|12 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 a|0, properties Partial order • 0|a a = 0, Prime number Division Algorithm • 1|a, Greatest common • ⇐⇒ divisor a|1 a = ±1, Definition Bezout • a|b ∧ b|a a = ±b Euclidean algorithm ⇐⇒ Extended Euclidean • Algorithm a|b −a|b a|−b Unique • a|b ∧ a|c =⇐⇒ a|(b + c), factorization into primes ⇐⇒ ⇐⇒ Some Lemmas • a|b = a|bc. 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 a|a, Partial order Prime number 2 a|b ∧ b|c = a|c, Division Algorithm
Greatest 3 a|b ∧ b|a = 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 ∈ P is a prime number if Prime number Division Algorithm • n > 1, Greatest common • m|n = m ∈ {1, n} 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 ∈ 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 |RHS| < b, so |LHS| < 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 ∈ . The greatest common divisor of a and b, c = gcd(a, b), is defined by Greatest Z common divisor 1 c|a ∧ c|b, Definition Bezout 2 If d|a ∧ d|b, then d ≤ c. Euclidean algorithm Extended Euclidean Algorithm If we restrict to P, the the last condition can be replaced with Unique 2’ If d|a ∧ d|b, then d|c. 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 Bezout’s theorem Theory, Lecture 1
Jan Snellman Theorem (Bezout) Divisibility Definition gcd Elementary Let d = (a, b). Then exists (not unique) x, y ∈ Z so that properties Partial order Prime number Division Algorithm ax + by = d.
Greatest common divisor Definition Proof. Bezout Euclidean algorithm min Extended Euclidean S = { ax + by x, y ∈ Z }, d = S ∩ P. If t ∈ S, then t = kd + r, 0 ≤ r < d. So Algorithm r = t − kd ∈ S ∩ . Minimiality of d, r < d gives r = 0. So d|t. Unique N factorization But a, b ∈ S, so d|a, d|b, and if ` another common divisor then a = `u, b = `v, and into primes Some Lemmas An importan property of primes d = ax + by = `ux + `vy = `(ux + vy) Euclid, again Fundamental theorem of arithmetic so `|d. Hence d is greatest common divisor. Exponent vectors Least common multiple
More about primes Sieve of Eratosthenes Primes in arithmetic progressions Number Etienne´ B´ezout Theory, Lecture 1
Jan Snellman
Divisibility Definition Elementary properties Partial order Prime number Division Algorithm
Greatest common divisor 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 Number Theory, Lecture 1
Jan Snellman
Divisibility Definition Elementary properties Lemma Partial order Prime number If a = kb + r then gcd(a, b) = gcd(b, r). Division Algorithm
Greatest common Proof. divisor Definition Bezout If c|a, c|b then c|r. Euclidean algorithm Extended Euclidean If c|b, c|r then c|a. 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 Number Extended Euclidean algorithm, example Theory, Lecture 1
Jan Snellman
Divisibility Definition Elementary properties Partial order Prime number Division Algorithm Greatest 6 = 1 ∗ 27 − 3 ∗ 7 common 27 = 3 ∗ 7 + 6 divisor 1 = 7 − 1 ∗ 6 Definition Bezout 7 = 1 ∗ 6 + 1 Euclidean algorithm = 7 − (27 − 3 ∗ 7) Extended Euclidean 6 = 6 ∗ 1 + 0 Algorithm = (−1) ∗ 27 + 4 ∗ 7 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 Number Xgcd Theory, Lecture 1
Jan Snellman
Divisibility Definition Elementary properties Partial order Algorithm Prime number Division Algorithm 1 Initialize: Set x = 1, y = 0, r = 0, s = 1. Greatest common divisor 2 Finished?: If b = 0, set d = a and terminate. Definition Bezout 3 Quotient and Remainder: Use Division algorithm to write a = qb + c with Euclidean algorithm Extended Euclidean 0 ≤ c < b. Algorithm Unique 4 Shift: Set (a, b, r, s, x, y) = (b, c, x − qr, y − qs, r, s) and go to Step 2. 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 Theory, Lecture 1
Jan Snellman Lemma Divisibility Definition gcd(an, bn) = |n| gcd(a, b). Elementary properties Partial order Prime number Proof Division Algorithm Greatest Assume a, b, n ∈ . Induct on a + b. Basis: a = b = 1, gcd(a, b) = 1, common P divisor gcd(an, bn) = n, OK. Definition Bezout Ind. step: a + b > 2, a ≥ b. Euclidean algorithm Extended Euclidean Algorithm a = kb + r, 0 ≤ r < b Unique factorization into primes Some Lemmas If k = 0, OK. Assume k > 0. 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 Then Divisibility Definition gcd gcd Elementary (a, b) = (b, r) properties Partial order gcd(an, bn) = gcd(bn, rn) Prime number Division Algorithm Greatest since common divisor an = kbn + rn, 0 ≤ rn < bn. Definition Bezout Euclidean algorithm But Extended Euclidean Algorithm b + r = b + (a − kb) = a − b(k − 1) ≤ a < a + b, Unique factorization into primes so ind. hyp. gives Some Lemmas n gcd(b, r) = gcd(bn, rn). An importan property of primes Euclid, again gcd gcd Fundamental But LHS = n (a, b), RHS = (an, bn). theorem of arithmetic Exponent vectors Least common multiple
More about primes Sieve of Eratosthenes Primes in arithmetic progressions Number Theory, Lecture 1
Jan Snellman Lemma Divisibility Definition gcd Elementary If a|bc and (a, b) = 1 then a|c. properties Partial order Prime number Division Algorithm Proof. Greatest common divisor Definition 1 = ax + by, Bezout Euclidean algorithm Extended Euclidean so Algorithm c = axc + byc. Unique factorization into primes Since a|RHS, a|c. 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 Definition Elementary properties Partial order Lemma Prime number Division Algorithm p prime, p|ab. Then p|a or p|b. Greatest common divisor Proof. Definition Bezout gcd Euclidean algorithm If p 6 |a then (p, a) = 1. Thus p|b by previous lemma. 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 Number Infinitude of primes Theory, Lecture 1
Jan Snellman
Divisibility Theorem (Euclides) Definition Elementary properties Ever n is a product of primes. There are infinitely many primes. Partial order Prime number Division Algorithm
Greatest Proof. common divisor 1 is regarded as the empty product. Ind on n. If n prime, OK. Otherwise, n = ab, Definition Bezout a, b < n. So a, b product of primes. Combine. Euclidean algorithm Extended Euclidean Suppose p1, p2,..., ps are known primes. Put Algorithm
Unique factorization N = p1p2 ··· ps + 1, into primes Some Lemmas An importan property of primes then N = kpi + 1 for all known primes, so no known prime divide N. But N is a Euclid, again Fundamental product of primes, so either prime, or product of unknown primes. 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 Elementary Example properties Partial order Prime number Division Algorithm Greatest 2 ∗ 3 ∗ 5 + 1 = 31 common divisor Definition 2 ∗ 3 ∗ 5 ∗ 7 + 1 = 211 Bezout Euclidean algorithm 2 ∗ 3 ∗ 5 ∗ 7 ∗ 11 ∗ 13 + 1 = 59 ∗ 509 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 Number Theory, Lecture 1
Jan Snellman
Divisibility Definition Elementary Example properties Partial order Prime number Division Algorithm Greatest 2 ∗ 3 ∗ 5 + 1 = 31 common divisor Definition 2 ∗ 3 ∗ 5 ∗ 7 + 1 = 211 Bezout Euclidean algorithm 2 ∗ 3 ∗ 5 ∗ 7 ∗ 11 ∗ 13 + 1 = 59 ∗ 509 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 Number Theory, Lecture 1
Jan Snellman
Divisibility Definition Elementary Example properties Partial order Prime number Division Algorithm Greatest 2 ∗ 3 ∗ 5 + 1 = 31 common divisor Definition 2 ∗ 3 ∗ 5 ∗ 7 + 1 = 211 Bezout Euclidean algorithm 2 ∗ 3 ∗ 5 ∗ 7 ∗ 11 ∗ 13 + 1 = 59 ∗ 509 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 Number Fundamental theorem of arithmetic Theory, Lecture 1
Jan Snellman
Divisibility Theorem Definition Elementary properties For any n ∈ P, can uniquely (up to reordering) write Partial order Prime number Division Algorithm n = p1p2 ··· ps , pi prime . Greatest common divisor Definition Bezout Proof. Euclidean algorithm Extended Euclidean Existence, Euclides. Uniqueness: suppose Algorithm
Unique factorization = = . into primes n p1p2 ··· ps q1q2 · qr Some Lemmas An importan property of primes Since p1|n, we have p1|q1q2 ··· qr , which by lemma yields p1|qj some qj , hence Euclid, again Fundamental p = q . Cancel and continue. theorem of 1 j arithmetic Exponent vectors Least common multiple
More about primes Sieve of Eratosthenes Primes in arithmetic progressions Number Exponent vectors Theory, Lecture 1
Jan Snellman • Number the primes in increasing order, p1 = 2,p2 = 3,p3 = 5, et cetera. aj Divisibility • Then n = j=1 pj , all but finitely many aj zero. Definition Elementary • Let v(n) = (a∞1, a2, a3,... ) be this integer sequence. properties Q Partial order • Then v(nm) = v(n) + v(m). Prime number Division Algorithm • Order componentwise, then n|m v(n) ≤ v(m). Greatest gcd min common • Have v( (n, m)) = (v(n), v(m)). divisor Definition ⇐⇒ Bezout Example Euclidean algorithm Extended Euclidean Algorithm Unique 2 2 factorization gcd(100, 130) = gcd(2 ∗ 5 , 2 ∗ 5 ∗ 13) into primes Some Lemmas min(2,1) min(2,1) min(0,1) An importan = 2 ∗ 5 ∗ 13 property of primes 1 1 0 Euclid, again = 2 ∗ 5 ∗ 13 Fundamental theorem of arithmetic = 10 Exponent vectors Least common multiple
More about primes Sieve of Eratosthenes Primes in arithmetic progressions Number Theory, Lecture 1 Definition
Jan Snellman • a, b ∈ Z Divisibility Definition • m = lcm(a, b) least common multiple if Elementary properties 1 m = ax = by (common multiple) Partial order Prime number 2 If n common multiple of a, b then m|n Division Algorithm
Greatest common divisor Lemma (Easy) Definition Bezout Euclidean algorithm • Extended Euclidean a, b ∈ P, c, d ∈ Z Algorithm max • lcm aj bj (aj ,bj ) Unique ( j pj , j pj ) = j pj factorization into primes • ab = gcd(a, b)lcm(a, b) Some Lemmas Q Q Q An importan property of primes • If a|c and b|c then lcm(a, b)|c Euclid, again Fundamental mod mod mod lcm theorem of • If c ≡ d a and c ≡ d b then c ≡ d (a, b) arithmetic Exponent vectors Least common multiple
More about primes Sieve of Eratosthenes Primes in arithmetic progressions Number Sieve of Eratosthenes Theory, Lecture 1 Jan Snellman Algorithm Divisibility Definition 1 Given N, find all primes ≤ N 5 P = P ∪ {pi } Elementary properties 2 X = [2, N], i = 1, P = ∅ Partial order √ Prime number 3 p = min(X ). 6 Division Algorithm i If pi ≥ N, then terminate, Greatest 4 Remove multiples of pi from X otherwise i = i + 1, goto 3. common divisor 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 Number Theory, Lecture 1
Jan Snellman
Divisibility Definition • Elementary Any number have remainder 0,1,2, or 3, when divided by 4 properties Partial order • Except for 2, all primes are odd Prime number Division Algorithm • Thus, primes > 2 are either of the form 4n + 1 or 4n + 3 Greatest common • divisor 4n + 3 = 4(n + 1) − 1 = 4m − 1. Definition Bezout Euclidean algorithm Theorem Extended Euclidean Algorithm There are infinitely many primes of the form 4m − 1. 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 Number Proof Theory, Lecture 1
Jan Snellman
Divisibility Proof. Definition Elementary Let q1,..., qr be the known such primes, put properties Partial order Prime number Division Algorithm N = 4q1q2 ··· qr − 1 Greatest common divisor Then N odd, not divisible by any qj . Factor N into primes: Definition Bezout Euclidean algorithm N = u1u2 ··· us Extended Euclidean Algorithm Unique If all u = 4m + 1 then factorization i i into primes Some Lemmas An importan N = (4m1 + 1)(4m2 + 1) ··· (4ms + 1) = 4m + 1, property of primes Euclid, again Fundamental theorem of a contradiction. So some uj = 4mj − 1, uj |N so uj 6∈ {q1,..., qr }, hence new. arithmetic Exponent vectors Least common multiple
More about primes Sieve of Eratosthenes Primes in arithmetic progressions Number Theory, Lecture 1
Jan Snellman
Divisibility Definition Elementary properties Theorem (Dirichlet) Partial order Prime number Division Algorithm a, b ∈ Z, gcd(a, b) = 1. Then aZ + b contains infinitely many primes. Greatest common divisor Example Definition Bezout Euclidean algorithm Obviously 6Z + 3 contains only one prime, 3, so condition necessary. 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 Number Dirichlet Theory, Lecture 1
Jan Snellman
Divisibility Definition Elementary properties Partial order Prime number Division Algorithm
Greatest common divisor 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