Dive into a comprehensive 16-hour course on Number Theory, exploring fundamental concepts and advanced topics. Begin with mathematical induction and progress through division algorithms, greatest common divisors, and the Euclidean algorithm. Examine prime numbers, modular arithmetic, and divisibility rules before delving into linear congruences and the Chinese Remainder Theorem. Study Euler's Theorem, Wilson's Theorem, and Hensel's Lemma, then investigate primitive roots and their applications. Explore quadratic residues, reciprocity, and forms, before concluding with an introduction to integer partitions, generating functions, and Ramanujan's Theta Functions. Master key theorems and conjectures while developing problem-solving skills in this in-depth exploration of number theory.
Overview
Syllabus
The foundation -- Number Theory 1.
Mathematical Induction -- Number Theory 2.
The division algorithm -- Number Theory 3.
The Greatest Common Divisor -- Number Theory 4.
The Euclidean Algorithm -- Number Theory 5.
Primes and Composites -- Number Theory 6.
Proofs and Conjectures involving primes -- Number Theory 7.
Modular Arithmetic -- Number Theory 8.
Divisibility Rules -- Number Theory 9.
Solving linear congruences -- Number Theory 10.
Chinese Remainder Theorem -- Number Theory 11.
Euler's Theorem -- Number Theory 12.
Euler's Totient Function -- Number Theory 13.
Wilson's Theorem -- Number Theory 14.
Hensel's Lemma -- Number Theory 15.
The order of an integer modulo n -- Number Theory 16.
Primitive Roots -- Number theory 17.
More about primitive roots -- Number Theory 18.
Applications of primitive roots -- Number Theory 19.
Indices (the discrete log) -- Number Theory 20.
Decimal Representations -- Number Theory 21.
Quadratic Residues -- Number Theory 22.
Quadratic Reciprocity proof -- Number Theory 23.
Quadratic Reciprocity Examples -- Number Theory 24.
Square roots mod p -- Number Theory 25.
Sums of squares -- Number Theory 26.
Quadratic Forms -- Number Theory 27.
Introduction to Integer Partitions -- Number Theory 28.
Generating Functions -- Number Theory 29.
How to use generating functions with integer partitions -- Number Theory 30.
Introduction to product-sum identities -- Number Theory 31.
Ramanujan's Theta Functions -- Number Theory 32.
Ramanujan's famous (mod 5) congruence -- Number Theory 33.
Difference 2 at distance 1 -- Number Theory Video 34.
Taught by
Michael Penn
