Number Theory

IGCSE

IB

USA

Singapore

Primary Year

Secondary Year

Junior College

Poly Technique

College

Course Content

UNIT1: Divisibility

1.1 Greatest common
1.2 Linear Diophantine equations
1.3 Primes and irreducibles
1.4 The fundamental theorem of arithmetic

UNIT 2: Multiplicative functions

UNIT 3: Modular arithmetic

UNIT 4: Primitive roots

UNIT 5: Quadratic residues

UNIT 6: Representation of integers by some quadratic forms

UNIT 7: Diophantine equations

UNIT 8: Continued fractions

UNIT 9: Diophantine approximations

UNIT 10: Quadratic number fields

UNIT 11: Chebyshev’s theorem

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Price : Free
Enrolled :0 Students
Duration :
Lessons :41 Lessons
Quiz :0 Quizzes
Skill Level :Intermediate
Category :College

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IGCSE

Primary Year

Secondary Year

Junior College

Poly Technique

Course Content

UNIT1: Divisibility

1.1 Greatest common

1.2 Linear Diophantine equations

1.3 Primes and irreducibles

1.4 The fundamental theorem of arithmetic

UNIT 2: Multiplicative functions

2.1 Basics

2.2 The Möbius inversion formula

UNIT 3: Modular arithmetic

3.1 Congruences

3.2 Wilson’s theorem

3.3 Fermat’s little theorem

3.4 The Chinese Remainder Theorem

3.5 Euler’s totient function

3.6 Euler’s theorem

UNIT 4: Primitive roots

4.1 The order of an element

4.2 Primitive roots modulo a prime

4.3 Primitive roots modulo prime powers

4.4 The structure of(Z/2kZ)×

UNIT 5: Quadratic residues

5.1 The Legendre symbol and Euler’s criterion

5.2 Gauss’s lemma

5.3 The quadratic reciprocity law

5.4 Composite moduli

UNIT 6: Representation of integers by some quadratic forms

6.1 Sums of two squares

6.2 Sums of four squares

UNIT 7: Diophantine equations

7.1 The Pythagorean equation

7.2 A special case of Fermat’s last theorem

7.3 Rational points on quadratic curves

UNIT 8: Continued fractions

8.1 Finite continued fractions

8.2 Representation of rational numbers by continued fractions

8.3 Infinite continued fractions

8.4 Periodic continued fractions

8.5 Pell’s equation

UNIT 9: Diophantine approximations

9.1 Dirichlet’s theorem

9.2 Better approximations

9.3 Transcendental numbers and Liouville’s theorem

9.4 Transcendence of e

UNIT 10: Quadratic number fields

10.1 Units

10.2 Gaussian integers

10.3 Fermat’s little theorem for Gaussian integers

10.4 Using Gaussian integers to solve Diophantine equations

UNIT 11: Chebyshev’s theorem

11.1 Basic estimates

11.2 Proof of Chebyshev’s theorem

11.3 On the prime number theorem

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