MU-207 Number Theory

Unit · year 2

MU-207 · Number Theory

Threads number24 lectures5 theorems

The deep structure of the integers, from unique factorisation to reciprocity.

PREREQUISITES

L01	Divisibility and the Division Algorithm —
L02	Greatest Common Divisors —
L03	The Euclidean Algorithm The Euclidean algorithm and Bézout's identity
L04	Bézout's Identity and Linear Diophantine Equations The Euclidean algorithm and Bézout's identity
L05	Primes and Euclid's Lemma The fundamental theorem of arithmetic
L06	The Fundamental Theorem of Arithmetic The fundamental theorem of arithmetic
L07	Consequences of Unique Factorisation The fundamental theorem of arithmetic
L08	The Distribution of Primes: First Estimates —
L09	Congruences and Residue Classes —
L10	Linear Congruences and Modular Inverses The Euclidean algorithm and Bézout's identity
L11	The Chinese Remainder Theorem Revisited —
L12	Euler's Totient Function Euler's theorem
L13	Euler's Theorem Euler's theorem
L14	Wilson's Theorem Wilson's theorem
L15	Primitive Roots and Primality Testing Euler's theorem
L16	Quadratic Residues The law of quadratic reciprocity
L17	The Legendre Symbol and Euler's Criterion The law of quadratic reciprocity
L18	Gauss's Lemma The law of quadratic reciprocity
L19	The Law of Quadratic Reciprocity The law of quadratic reciprocity
L20	Applications of Reciprocity The law of quadratic reciprocity
L21	Sums of Two Squares —
L22	Continued Fractions and Pell's Equation —
L23	Public-Key Cryptography: RSA Euler's theorem
L24	Synthesis: Arithmetic as Structure The fundamental theorem of arithmetic, The law of quadratic reciprocity

The gcd is an integer combination of its arguments.

Every integer factors uniquely into primes.

a^φ(n) ≡ 1 (mod n) for a coprime to n.

(p−1)! ≡ −1 (mod p) exactly when p is prime.

A reciprocal relationship between two primes being squares mod each other.