Text (required): I. Niven, H. S. Zuckerman and H. L. Montgomery, An Introduction to the Theory of Numbers, 5th edition, John Wiley & Sons, 1991.
This syllabus is tentative. It will be superseded by later versions as the course evolves. (Oct. 17, 2009 version)
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# | date | sections in Niven and Zuckerman | HW due dates | problems |
1 | 9/9 | History: Babylonian/Greek | ||
2 | 9/11 | History: through Gauss, Sec. 1.2 induction, start theorem 1.2 | ||
3 | 9/14 | Sec. 1.2 GCD; state Euclidean algorithm | ||
4 | 9/16 | Sec.1.3 GCD, LCM, start fundamental theorem of arithmetic | ||
5 | 9/18 | Sec. 1.3 Fundamental theorem of arithmetic; infinity of primes, and primes of form 4k+3, \sum 1/p > loglog x -C | HW#1 due Fri. 9/18 | |
6 | 9/21 | Sec. 1.4 Binomial coefficients, Sec. 2.1 congruences | ||
7 | 9/23 | Facts on primes, Sect. 2.1, congruences, Euler's Thm. [ Lecture: DeBacker] | ||
8 | 9/25 | Fermat's little thm (Euler thm) Sec. 2.1 (Wilson's thm.), 2.2 (linear congruences), 2.3 (Chinese remainder theorem) [Lecture: Prasanna] | ||
9 | 9/28 | Sec. 2.4 computing powers (mod m), 2.3 Chinese remainder theorem =Lagrange interpolation, Sec. 2.1 (sum of two squares), | HW#2 due Tues. 9/29 | |
10 | 9/30 | Sec. 2.3 (multiplicative functions) Sect. 4.2 (Theorem 4.4) multiplicative function | ||
11 | 10/2 | Sec 2.5 Public key cryptography, computing exponents | ||
12 | 10/5 | Facts on factoring and primality testing, Sect. 2.7 congruences (mod p), Lagrange's theorem | ||
13 | 10/7 | Sect. 2.7 Factoring x^n-1,
Wolstenholme's theorem Sect. 2.6 prime power moduli: Taylor expansion, binomial theorem, Hensel's lemma, nonsingular case |
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14 | 10/9 | Sect. 2.6 Hensel's lemma: singular case; p-adic numbers, absolute values | HW#3 due Fri. Oct. 9 | |
15 | 10/12 | p-adic numbers-continued; Ostrowski theorem statement, | ||
16 | 10/14 | Sect. 2.8 Primitive roots (mod p). | ||
17 | 10/16 | Midterm Exam 1 (in class) | Covers Chapters 1 and 2 through 2.7: | |
10/19 | Fall Study Break | |||
18 | 10/21 | Sect. 2.8
Primitive roots (general case); quadratic residues primality testing |
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19 | 10/23 | Sect. 2.9 Applications of
primitive roots; power residues |
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20 | 10/26 | 3.1 Quadratic Residues, Gauss Lemma , | HW#4 due Mon. Oct. 26 | |
21 | 10/28 | 3.2 quadratic reciprocity | ||
22 | 10/30 | 3.2/3.3 Quadratic reciprocity, Jacobi symbol | ||
23 | 11/2 | 3.3 Counting solutions to two-varible congruences | ||
23 | 11/4 | 3.4 Binary quadratic forms | HW#5 due Wed. Nov. 4 | |
24 | 11/6 | 3.5 Equivalence of binary quadratic forms; class number | ||
25 | 11/9 | 3.5 Reduction of binary forms | ||
26 | 11/11 | 3.7 Positive definite forms,examples | ||
27 | 11/13 | 3.7 Positive definite forms, unique reduction | ||
28 | 11/16 | 3.7 Positive def. forms, automorphs, representations | HW#6 due Mon. Nov. 16 | |
29 | 11/18 | 4.2 Arithmetic Functions | ||
30 | 11/20 | 4.2, 4.3 Arithmetic functions/ Mobius inversion | ||
31 | 11/23 | Midterm 2 (Mon. Nov. 23) | Covers Chapter 2.8: Chap. 3 [Does not cover Chap. 4] | |
32 | 11/25 | 8.3 Estimates of arithmetical functions, d(n) | ||
11/27 | Thanksgiving | |||
33 | 11/30 | 8.3 Estimates of Arithmetical functions, squarefree, sums of two squares | ||
34 | 12/2 | 7.1-7.2 Diophantine approximation, continued fractions | HW#7 due | |
35 | 12/4 | 7.1, 7.2 Continued fractions | ||
36 | 12/7 | 7.7 Periodic continued fractions | ||
37 | 12/9 | 7.7, 7.8 Diophantine equations; Pell's equation | HW#8 due | Take-home Final Exam handed out : |
38 | 12/11 | Liouville's theorem; irrationality | ||
39 | 12/14 | Diophantine equations; even and odd Perfect numbers | Take Home Final Due 5pm |