| Number Theory Schedule | |||||
| rev. 11/1; forecast is subject to change | |||||
| week | topics… | read… | hw due | ||
| 1 | 5-Sep | Tu | Euclidean algorithm for d=gcd(a,b); expressing d as lin. comb. of a,b; If a|bc and (a,b)=1 then a|c. | 3.3 (Or Euclid, Book 7, Props. 1-2), 3.4 | |
| 7-Sep | Th | Axioms including well-ordering; induction; even perfect numbers | App. A, 1.3 | 1 | |
| 2 | 12-Sep | Tu | Primes; distribution; recognition by trial division, seives, and Fermat's little theorem p|a^p-a | 3.1, 3.2 (Fermat is in Sec. 6.1) | 2 |
| 14-Sep | Th | Unique factorization | 3.5 | 3 | |
| 3 | 19-Sep | Tu | Congruences; 1007 isn't a sum of two squares; exponentiation mod m | 4.1 (Or Gauss 1801, Book 1, Sec. 1) | 4 (prove fta) |
| 21-Sep | Th | Solving linear congruences ax=c(mod m) when a has a multiplicative inverse; Wilson's theorem (p-1)!=-1(mod p) | 4.2 (Wilson is in Sec. 6.1) | 5 | |
| 4 | 26-Sep | Tu | More congruences; partial orderings? | 4.2 | 6 |
| 28-Sep | Th | Chinese Remainder Theorem | 4.3 | 7 | |
| 5 | 3-Oct | Tu | Understanding the multiplicative structure of Z-mod-p (primitive roots and powers) | 8 | |
| 5-Oct | Th | Check digits; ISBN | Sec. 5.5 | 9; Exam 1 distributed | |
| 6 | 10-Oct | Tu | Multiplicative functions: tau, sigma, phi | Sec. 7.1-2, part of 3 | Exam 1 due |
| 12-Oct | Th | More multiplicative functions: perfect numbers, convolutions, Mobius inversion | Sec. 7.1 - ex. 37-42 and note preceding 37; also 7.4 | 10 | |
| October Holiday | |||||
| 7 | 24-Oct | Tu | More Mobius: the necklace problem; Euler's theorem | Sec. 6.3 | 11 |
| 26-Oct | Th | Lagrange on polynomial having at most n roots; Existence of primitive roots | Roughly, 9.1-9.2 (Lagrange is Thm. 9.6) | 11 again | |
| 8 | 31-Oct | Tu | Quadratic residues: Euler criterion, -1, Gauss's lemma | Sec. 11.1 | |
| 2-Nov | Th | When 2 is a qr; quadratic reciprocity; evaluating all Legendre symbols | Sec. 11.2 | ||
| 9 | 7-Nov | Tu | Diophantine Equations: Pythagorean triples, Fermat's Last Theorem | Sec. 13.1, 13.2 | |
| 9-Nov | Th | Sums of squares (consolidate two squares; three squares; four squares) | Sec. 13.3 | ||
| 10 | 14-Nov | Tu | Recognizing primes: pseudoprimes, Rabin test | ||
| 16-Nov | Th | "PRIMES is in P" | |||
| 11 | 21-Nov | Tu | RSA encryption | ||
| Thanksgiving Break | |||||
| 12 | 28-Nov | Tu | Factoring | ||
| 30-Nov | 0 | Factoring: Quadratic seive | |||
| 13 | 5-Dec | Tu | Analytic number theory: Dirichlet theorem, the Riemannn hypothesis, and the prime number theorem | ||
| 7-Dec | 0 | More D., R., and PNT | |||
| 14 | 12-Dec | Tu | Class party. Oh, where are the zeros of zeta of s? | ||
| December 15 - 22 : Final Exams | |||||