Math 40520 Theory of Numbers (Fall 2018)
Undergraduate number theory
| Date | Lecture | Topics | Sage |
| Wed 08/22 | 1 | Division with remainder 1.1.11, the Euclidean algorithm 1.1.10, Bezout's identity 2.3.1 | |
| Fri 08/24 | 2 | Prime numbers 1.2.1, unique factorization 1.1.20 | |
| Mon 08/27 | 3 | Fermat primes, Mersenne primes 1.2.3, prime factorizations of factorials | sage |
| Wed 08/29 | 4 | Factorials and base p, the number of primes 1.2.5 | |
| Fri 08/31 | 5 | The prime number theorem | sage |
| Mon 09/03 | 6 | Modular arithmetic 2.1 and 2.3.2, invertible mod n 2.1.13, exponentiation 2.3.2 | |
| Wed 09/05 | 7 | Solving diophantine equations using congruences | sage |
| Fri 09/07 | 8 | Infinitely many prime which are 3 mod 4 1.2.5, Chinese Remainder Theorem 2.2.2 | |
| Mon 09/10 | 9 | Explicit CRT 2.2.3 and the Euler function 2.2.1 | sage |
| Wed 09/12 | 10 | Fermat's little theorem, Euler's theorem 2.1.2, multiplicative order 2.1.16 | sage |
| Fri 09/14 | 11 | Multiplicative orders | |
| Mon 09/17 | 12 | Primitive roots mod p 2.5 | sage |
| Wed 09/19 | 13 | Wilson's theorem and the existence of primitive roots mod p | |
| Fri 09/21 | 14 | The structure of units mod powers of primes | |
| Mon 09/24 | 15 | Polynomials mod n and the Chinese Remainder Theorem | |
| Wed 09/26 | 16 | Counting roots mod p | sage |
| Fri 09/28 | 17 | Applications of Hensel's lemma | |
| Mon 10/01 | 18 | Proof of Hensel's lemma | |
| Wed 10/03 | 19 | Hensel's lifting lemma in the non-unique case | |
| Fri 10/05 | 20 | Public key cryptography | |
| Mon 10/08 | 21 | Elgamal | |
| Wed 10/10 | 22 | RSA and Rabin | sage |
| Fri 10/12 | 23 | Zero knowledge proofs | |
| Mon 10/22 | 24 | Quadratic residues and Gauss' Lemma 4.3.1 | |
| Wed 10/24 | 25 | Proof of Gauss' Lemma | |
| Fri 10/26 | 26 | Quadratic reciprocity, randomness of Legendre symbols | sage |
| Mon 10/29 | 27 | Computing Legendre symbols, sums of Legendre symbols | sage |
| Wed 10/31 | 28 | Sums of Legendre symbols | |
| Fri 11/02 | 29 | Diophantine equations mod p | |
| Mon 11/05 | 30 | Elliptic curves mod p | |
| Wed 11/07 | 31 | Hyperboloids mod p | sage |
| Fri 11/09 | 32 | Proof of quadratic reciprocity | |
| Mon 11/12 | 33 | Finish proof of quadratic reciprocity | |
| Wed 11/14 | 34 | Primality testing | |
| Fri 11/16 | 35 | Primality testing | sage |
| Mon 11/19 | 36 | Binomial coefficients mod primes: Lucas' theorem | |
| Mon 11/26 | 37 | Binomial coefficients mod primes continued | |
| Wed 11/28 | 38 | Dirichlet series | |
| Fri 11/30 | 39 | Products of Dirichlet series | |
| Mon 12/03 | 40 | The Mobius inversion formula | |
| Wed 12/05 | 41 |
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