|Fall 2020 MAT 311: Number theory|
|Schedule||MW 2:40-4:00pm Physics P-113|
|Office hours||F 9-11am in Math Tower 4-118, W 6-7pm in Math Learning Center|
|Description||Congruences, quadratic residues, quadratic forms, continued fractions, Diophantine equations, number- theoretical functions, and properties of prime numbers.|
|Prerequisites||C or higher in MAT 312 or 313 or 318; C or higher in MAT 200 or permission of instructor|
|Textbook||Niven, Zuckerman and Montgomery. Introduction to the theory of numbers. John Wiley and Sons (1991).|
|Homework||Weekly problem sets will be assigned, and collected in class on Monday. Late homework is not accepted, but under documented extenuating circumstances the grade may be dropped. Your lowest homework grade will be dropped at the end of the class.|
|Grading||Homework: 20%, Midterm I: 20%, Midterm II: 20%, Final: 40%.|
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Syllabus/schedule (subject to change)
|Mon 8/24||1.||Divisibility, primes, the binomial theorem||1.1-1.4, 2.1-2.3|
|Wed 8/26||2.||Congruences, Chinese remainder theorem, Slides||HW Due 8/31: p.17 #2, 7, 46, p.29 #5, 37, p.40 #1, 3, 10, p.56 #15, 18, 47, 57, p.62 #5, p.71 #3, 19, 21|
|Mon 8/31||3.||Cryptography, Slides||2.4-2.11|
|Wed 9/2||4.||Primitive roots, groups, rings and field, Boards||HW Due 9/9: p.82 #4, 13, p.91 #3, 9, p.96 #1, 4, p.106 #1, 22, 32, p.114 #2, 3, p.119 #2, p.126 #20, 22, 23|
|Mon 9/7||No class - Labor day||3.1-3.3|
|Wed 9/9||5.||Quadratic reciprocity, Boards||HW Due 9/14: p.135 #8, 11, 24, p.140 #1, 7, 24, p.147 #1, 5, 17|
|Mon 9/14||6.||Binary quadratic forms, reduction theory, Boards||3.4-3.7, 4.1-4.3|
|Wed 9/16||7.||Unique factorization in the Gaussian integers, Boards||HW Due 9/21: p.154 #4, 5, p.162 #2, 13, 15, 16, p.169 #2, 12, p.176 #1, 4|
|Mon 9/21||8.||The class number formula, Boards||4.4-4.5|
|Wed 9/23||Midterm 1 Practice Midterm, Solutions||Midterm 1 solutions|
|Mon 9/28||9.||Arithmetic functions, Mobius inversion, Boards||5.1-5.6|
|Wed 9/30||10.||Linear Diophantine equations, Ternary quadratic forms, Boards||HW Due 10/5: p.184 #15, 20, p.191 #20, p.195 #7, 19, 24, 28, p.204 #3, 6, 7, 15, 25, p.210 #2, 4, 9, 10, 14|
|Mon 10/5||11.||Diophantine Equations, Boards||5.7-5.9, 6.1-6.2|
|Wed 10/7||12.||Elliptic curves, Boards||HW Due 10/12: p.218 #2, 5, p.229 #1, 6, 10, p.233 #10, 14, p.239 #14, 15, p.248 #1, 3, p.260 #1, 7, 14|
|Mon 10/12||13.||Farey fractions, rational approximation Boards||HW Due 10/12: 6.3-6.4, 7.1-7.3|
|Wed 10/14||14.||Geometry of numbers Boards||HW Due 10/19: p.278 #3, 10, p.287 #3, 6, p.300 #1, 4, 5, 9, p.307 #1, 2, 5|
|Mon 10/19||15.||Euclidean algorithm, continued fractions, Boards||7.4-7.9|
|Wed 10/21||16.||Irrational numbers, best approximation, Boards||HW Due 10/28: p.311 #3, 8, 9, p.319 #7, 12, 13, 14, p.327 #1, p.329 #1, p.333 #3|
|Mon 10/26||17.||Pell's equation, Boards||8.1-8.2|
|Wed 10/28||Midterm 2 Practice Midterm 2, Solutions, Midterm 2 solutions||HW Due 11/2: p.336 #1, 2, p.340 #3, 4, p.344 #4, 5, p.351 #1, p.356 #1, 2, 6|
|Mon 11/2||18.||Prime number theory Boards||8.3-8.4, 9.1-9.3|
|Wed 11/4||19.||Dirichlet series Boards||HW Due 11/9: p.373 #1, 2, 6, 19, 20, p.387 #4, 6, 7, 23, 24|
|Mon 11/9||20.||Primes in arithmetic progression, Slides||9.4-9.8|
|Wed 11/11||21.||Algebraic numbers Boards||HW Due 11/16: p.398 #3, 5, 16, p.405 #5, 6, p.414 #5, 7, 9, p.419 #1, 2, 3, p.423 #2, 3|
|Mon 11/16||22.||Factorization into prime ideals Boards||9.9-9.10, 10.1-10.3|
|Wed 11/18||23.||Cubic case of Fermat's last theorem Boards||HW Due 11/30: p.425 #2, 3, p.427 #1, 2, 3, 4, 5, p.429 #1, p.431 #2, 3, 5, p.432 #1, 4|
|Mon 11/23||No class - Thanksgiving|
|Wed 11/25||No class - Thanksgiving|
|Mon 11/30||24.||Partitions, formal power series||10.4-10.6, 11.1-11.2|
|Wed 12/2||25.||Euler's formula, bounds on the partition number||HW Due 12/7 (optional): p.440 #1, 2, 6, p.445 #1, p.447 #1, p.450 #1, 2, p.457 #1, 2, 3|
|Mon 12/7||26.||Asymptotic density|
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