https://wiki.math.wisc.edu/api.php?action=feedcontributions&user=Sjohnson&feedformat=atomUW-Math Wiki - User contributions [en]2022-10-05T08:56:41ZUser contributionsMediaWiki 1.35.6https://wiki.math.wisc.edu/index.php?title=Math847&diff=3102Math8472011-11-16T18:57:06Z<p>Sjohnson: </p>
<hr />
<div>Here is the sign-up for lecture notes for Math 847. If you need to switch a spot, please find someone to trade so as to not leave any spot empty. Everyone will need to eventually sign-up for three days, ideally one from the beginning, one from the middle, and one from the end of class. Feel free to edit this wiki as needed.<br />
<br />
Friday September 1 Andrew Bridy<br />
<br />
Wednesday September 7 Silas Johnson<br />
<br />
Friday September 9 Daniel Ross<br />
<br />
Monday September 12 Nathan Clement<br />
<br />
Wednesday September 14 Ahmet Kabakulak<br />
<br />
Friday September 16 Yongqiang Zhao<br />
<br />
Monday September 19 Lalit <br />
<br />
Wednesday September 21 Yueke Hu<br />
<br />
Friday September 23 Qian You<br />
<br />
Monday September 26 Ting-Ting Nan<br />
<br />
Wednesday September 28 Peng Yu <br />
<br />
Friday September 30 Derek <br />
<br />
Monday October 3 Christelle <br />
<br />
Wednesday October 5 Mimansa <br />
<br />
<br />
Monday October 10 Huanyu Wen<br />
<br />
Wednesday October 12 luanlei zhao<br />
<br />
Friday October 14 David <br />
<br />
Monday October 17 Qian You<br />
<br />
Wednesday October 19 Rohit Nagpal<br />
<br />
Friday October 21 Ed<br />
<br />
Mon Oct 24 Ahmet<br />
<br />
Wed Oct 26 Lalit<br />
<br />
Fri Oct 28 Daniel<br />
<br />
Mon Oct 31 Andrew<br />
<br />
Wed Nov 2 Yueke<br />
<br />
Fri Nov 4 Peng<br />
<br />
There will be no class Nov 7-11<br />
<br />
Mon Nov 14 Luanlei Zhao<br />
<br />
Wednesday November 16 Huanyu<br />
<br />
Fri Nov 18 Qian<br />
<br />
Mon Nov 21 Ting-Ting<br />
<br />
Wed Nov 23 Silas<br />
<br />
Mon Nov 28 Mimansa<br />
<br />
<br />
Mon Dec 5 Ahmet Kabakulak<br />
<br />
Wed Dec 7 Luanlei Zhao<br />
<br />
Fri Dec 9 Peng<br />
<br />
Mon Dec 12 Mimansa <br />
<br />
Wed Dec 14 Daniel<br />
<br />
<br />
If you don't get a third note slot, you should sign up below for two(ish) pages of examples related to your favorite part of class.<br />
<br />
____________________<br />
_____________________<br />
____________________</div>Sjohnsonhttps://wiki.math.wisc.edu/index.php?title=NTSGrad&diff=1441NTSGrad2011-01-18T13:24:41Z<p>Sjohnson: /* Spring 2011 Semester */</p>
<hr />
<div>= Number Theory Graduate Student Seminar, University of Wisconsin-Madison =<br />
<br />
<br />
*'''When:''' Tuesdays at 2:30pm<br />
*'''Where:''' Van Vleck Hall B105 (or possibly B129)<br />
<br />
<br><br />
<br />
The purpose of this seminar is to have a talk on each Tuesday by a graduate student to<br />
help orient ourselves for the [[NTS|Number Theory Seminar]] talk on the following Thursday.<br />
These talks should be aimed at beginning graduate students, and should try to <br />
explain some of the background, terminology, and ideas for the Thursday talk. <br />
<br />
== Spring 2011 Semester ==<br />
<br />
<center><br />
<br />
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"<br />
|-<br />
| bgcolor="#D0D0D0" width="300" align="center"|'''Date'''<br />
| bgcolor="#F0A0A0" width="300" align="center"|'''Speaker'''<br />
| bgcolor="#BCD2EE" width="300" align="center"|'''Title (click to see abstract)'''<br />
|-<br />
| bgcolor="#E0E0E0"| January 18 (Tue.)<br />
| bgcolor="#F0B0B0"| <br> <br />
| bgcolor="#BCE2FE"|[[NTS/Abstracts# | <font color="black"><em>Organizational meeting</em></font>]]<br />
|-<br />
| bgcolor="#E0E0E0"| January 25 (Tue.)<br />
| bgcolor="#F0B0B0"| <br> Silas Johnson<br />
| bgcolor="#BCE2FE"|[[NTS/Abstracts# | <font color="black"><em>TBA</em></font>]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 1 (Tue.)<br />
| bgcolor="#F0B0B0"| <br> <br />
| bgcolor="#BCE2FE"|[[NTS/Abstracts# | <font color="black"><em>TBA</em></font>]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 8 (Tue.)<br />
| bgcolor="#F0B0B0"| <br> Evan Dummit<br />
| bgcolor="#BCE2FE"|[[NTS/Abstracts | <font color="black"><em>An introduction to Stark's conjecture</em></font>]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 15 (Tue.)<br />
| bgcolor="#F0B0B0"| <br> <br />
| bgcolor="#BCE2FE"|[[NTS/Abstracts | <font color="black"><em>TBA</em></font>]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 22 (Tue.)<br />
| bgcolor="#F0B0B0"| <br> <br />
| bgcolor="#BCE2FE"|[[NTS/Abstracts | <font color="black"><em>TBA</em></font>]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 5 (Tue.)<br />
| bgcolor="#F0B0B0"|<br> <br />
| bgcolor="#BCE2FE"|[[NTS/Abstracts | <font color="black"><em>TBA</em></font>]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 8 (Tue.)<br />
| bgcolor="#F0B0B0"| <br> <br />
| bgcolor="#BCE2FE"|[[NTS/Abstracts | <font color="black"><em>TBA</em></font>]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 15 (Tue.)<br />
| bgcolor="#F0B0B0"| No seminar (spring break)<br> <br />
| bgcolor="#BCE2FE"|[[NTS/Abstracts | <font color="black"><em>TBA</em></font>]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 22 (Tue.)<br />
| bgcolor="#F0B0B0"| <br> <br />
| bgcolor="#BCE2FE"|[[NTS/Abstracts | <font color="black"><em>TBA</em></font>]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 29 (Tue.)<br />
| bgcolor="#F0B0B0"| <br> <br />
| bgcolor="#BCE2FE"|[[NTS/Abstracts | <font color="black"><em>TBA</em></font>]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 5 (Tue.)<br />
| bgcolor="#F0B0B0"| <br> <br />
| bgcolor="#BCE2FE"|[[NTS/Abstracts | <font color="black"><em>TBA</em></font>]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 12 (Tue.)<br />
| bgcolor="#F0B0B0"| <br> <br />
| bgcolor="#BCE2FE"|[[NTS/Abstracts | <font color="black"><em>TBA</em></font>]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 19 (Tue.)<br />
| bgcolor="#F0B0B0"| <br> <br />
| bgcolor="#BCE2FE"|[[NTS/Abstracts | <font color="black"><em>TBA</em></font>]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 21 (Tue.)<br />
| bgcolor="#F0B0B0"| <br> <br />
| bgcolor="#BCE2FE"|[[NTS/Abstracts | <font color="black"><em>TBA</em></font>]]<br />
|}<br />
</center><br />
<br />
<br><br />
<br />
== Organizer contact information ==<br />
[http://www.math.wisc.edu/~brownda/ David Brown:]<br />
<br />
[http://www.math.wisc.edu/~cais/ Bryden Cais:]<br />
<br />
<br><br />
<br />
<br />
----<br />
The Fall 2010 NTS Grad page has moved [http://www.math.wisc.edu/wiki/index.php/NTSGrad_Fall_2010 here:].<br />
----<br />
Return to the [[NTS|Number Theory Seminar Page]]<br />
<br />
Return to the [[Algebra|Algebra Group Page]]</div>Sjohnsonhttps://wiki.math.wisc.edu/index.php?title=Math_567_--_Elementary_Number_Theory&diff=885Math 567 -- Elementary Number Theory2010-09-16T16:28:54Z<p>Sjohnson: </p>
<hr />
<div>'''MATH 567<br />
<br />
Elementary Number Theory'''<br />
<br />
MWF 1:20-2:10, Van Vleck B119<br />
<br />
'''Professor:''' [http://www.math.wisc.edu/~ellenber/ Jordan Ellenberg] (ellenber@math.wisc.edu)<br />
''Office Hours:'' Weds, 2:30-3:30, Van Vleck 323. <br />
<br />
'''Grader:''' [http://www.math.wisc.edu/~sjohnson/ Silas Johnson] (snjohnson3@wisc.edu)<br />
''Office Hours:'' Thurs, 1:15-2:15, Van Vleck 522. [http://math.wisc.edu/~sjohnson/567F10 Silas's page], with problem set solutions.<br />
<br />
Math 567 is a course in elementary number theory, aimed at undergraduates majoring in math or other quantitative disciplines. A general familiarity with abstract algebra at the level of Math 541 will be assumed, but students who haven't taken 541 are welcome to attend if they're willing to play a little catchup. We will be using William Stein's new (and cheap) textbook [http://www.amazon.com/Elementary-Number-Theory-Computational-Undergraduate/dp/0387855246 Elementary Number Theory: Primes, Congruences, and Secrets], which emphasizes computational approaches to the subject. If you don't need a physical copy of the book, [http://www.williamstein.org/ent/ it is available as a free legal .pdf.] We will be using the (free, public-domain) mathematical software [http://www.sagemath.org/ SAGE], developed largely by Stein, as an integral component of our coursework. There is a [http://www.sagemath.org/pdf/SageTutorial.pdf useful online tutorial.] You can download SAGE to your own computer or [http://www.sagenb.org use it online].<br />
<br />
Topics include some subset of, but are not limited to: Divisibility, the Euclidean algorithm and the GCD, linear Diophantine equations, prime numbers and uniqueness of factorization. Congruences, Chinese remainder theorem, Fermat's "little" theorem, Wilson's theorem, Euler's theorem and totient function, the RSA cryptosystem. Number-theoretic functions, multiplicative functions, Möbius inversion. Primitive roots and indices. Quadratic reciprocity and the Legendre symbol. Perfect numbers, Mersenne primes, Fermat primes. Pythagorean triples, Special cases of Fermat's "last" theorem. Fibonacci numbers. Continued fractions. Distribution of primes, discussion of prime number theorem. Primality testing and factoring algorithms.<br />
<br />
<br />
'''Course Policies:''' Homework will be due on Fridays. It can be turned in late only with ''advance'' permission from your grader. It is acceptable to use calculators and computers on homework (indeed, some of it will require a computer) but calculators are not allowed during exams. You are encouraged to work together on homework, but writeups must be done individually.<br />
<br />
Many of the problems in this course will ask you to prove things. I expect proofs to be written in English sentences; the proofs in Stein's book are a good model for the level of verbosity I am looking for. <br />
<br />
'''Grading:''' The grade in Math 567 will be composed of 40% homework, 20% each of three midterms. The last midterm will be take-home, and will be due on the last day of class. ''There will be no final exam in Math 567''.<br />
<br />
'''Syllabus:''' <br />
(This may change as we see what pace works well for the course. All section numbers refer to Stein's book.)<br />
<br />
* Sep 3-10: Prime numbers, prime factorizations, Euclidean algorithm and GCD (1.1-1.2)<br />
* Sep 13-17: The integers mod n, Euler's theorem, the phi function (2.1-2.2)<br />
* Sep 20-24: Modular exponentiation, primality testing, and primitive roots (2.4-2.5)<br />
* Sep 27-Oct 1: Public-key cryptography and RSA (3.1-3.4)<br />
* Oct 4 - 8: Algebraic numbers <br />
* Oct 6: ''First midterm exam''<br />
* Oct 11-15: Quadratic reciprocity (4.1-4.4)<br />
* Oct 18-22: Finite and infinite continued fractions (5.1-5.3)<br />
* Oct 25-29: Continued fractions and diophantine approximation (5.4-5.5)<br />
* Nov 1-5: Diophantine equations I: Pell's equation and Lagrange's theorem<br />
* Nov 8-12: Elliptic curves (6.1-6.2)<br />
* Nov 10: ''Second midterm exam''<br />
* Nov 15-19: Applications of elliptic curves (6.3-6.4)<br />
* Nov 22-Dec 3: Diophantine equations II: Fermat, generalized Fermat, and probabilistic methods<br />
* Dec 6-15: advanced topic TBD: maybe a look at the Sato-Tate conjecture?<br />
<br />
<br />
<br />
'''Homework:'''<br />
Homework is due at the beginning of class on the specified Friday. Typing your homework is not a requirement, but if you don't already know LaTeX I highly recommend that you learn it and use it to typeset your homework. I will sometimes assign extra problems, which I will e-mail to the class list and include here.<br />
<br />
* Sep 13 (note this is Monday, not Friday!): 1.1, 1.3, 1.5, 1.7 (use SAGE), 1.8, 1.14.<br />
Problem A: Use SAGE to compute the number of x in [1..N] such that x^2 + 1 is prime, for N = 100, N = 1000, and N = 10000. Let f(N) be the number of such N. <br />
<br />
a) Can you formulate a conjecture about the relationship between f(N) and N/log N? <br />
<br />
b) What if x^2 + 1 is replaced with x^2 + 2? Can you explain why x^2 + 2 appears less likely to be prime? (Hint: consider x mod 3.)<br />
<br />
c) Prove that f(N) is at most (1/2)N+1. (Hint: consider x mod 2.) <br />
<br />
d) Give as good an upper bound as you can for f(N).<br />
<br />
Note that, despite the evident regularities you'll observe in this problem, we do not even know whether there are infinitely many primes of the form x^2 + 1! You would become very famous if you proved this.<br />
<br />
* Sep 17: 2.6 (the formulation of numerical evidence should be done by Sage if you've got Sage working, and by calculator if not; you can use an online tool like [http://primes.utm.edu/curios/includes/primetest.php this] to test whether a number is prime.) 2.8,2.9,2.11,2.12,2.14,2.19</div>Sjohnsonhttps://wiki.math.wisc.edu/index.php?title=Math_567_--_Elementary_Number_Theory&diff=884Math 567 -- Elementary Number Theory2010-09-16T16:28:40Z<p>Sjohnson: </p>
<hr />
<div>'''MATH 567<br />
<br />
Elementary Number Theory'''<br />
<br />
MWF 1:20-2:10, Van Vleck B119<br />
<br />
'''Professor:''' [http://www.math.wisc.edu/~ellenber/ Jordan Ellenberg] (ellenber@math.wisc.edu)<br />
''Office Hours:'' Weds, 2:30-3:30, Van Vleck 323. <br />
<br />
'''Grader:''' [http://www.math.wisc.edu/~sjohnson/ Silas Johnson] (snjohnson3@wisc.edu)<br />
''Office Hours:'' Thurs, 1:15-2:15, Van Vleck 522. [math.wisc.edu/~sjohnson/567F10 Silas's page], with problem set solutions.<br />
<br />
Math 567 is a course in elementary number theory, aimed at undergraduates majoring in math or other quantitative disciplines. A general familiarity with abstract algebra at the level of Math 541 will be assumed, but students who haven't taken 541 are welcome to attend if they're willing to play a little catchup. We will be using William Stein's new (and cheap) textbook [http://www.amazon.com/Elementary-Number-Theory-Computational-Undergraduate/dp/0387855246 Elementary Number Theory: Primes, Congruences, and Secrets], which emphasizes computational approaches to the subject. If you don't need a physical copy of the book, [http://www.williamstein.org/ent/ it is available as a free legal .pdf.] We will be using the (free, public-domain) mathematical software [http://www.sagemath.org/ SAGE], developed largely by Stein, as an integral component of our coursework. There is a [http://www.sagemath.org/pdf/SageTutorial.pdf useful online tutorial.] You can download SAGE to your own computer or [http://www.sagenb.org use it online].<br />
<br />
Topics include some subset of, but are not limited to: Divisibility, the Euclidean algorithm and the GCD, linear Diophantine equations, prime numbers and uniqueness of factorization. Congruences, Chinese remainder theorem, Fermat's "little" theorem, Wilson's theorem, Euler's theorem and totient function, the RSA cryptosystem. Number-theoretic functions, multiplicative functions, Möbius inversion. Primitive roots and indices. Quadratic reciprocity and the Legendre symbol. Perfect numbers, Mersenne primes, Fermat primes. Pythagorean triples, Special cases of Fermat's "last" theorem. Fibonacci numbers. Continued fractions. Distribution of primes, discussion of prime number theorem. Primality testing and factoring algorithms.<br />
<br />
<br />
'''Course Policies:''' Homework will be due on Fridays. It can be turned in late only with ''advance'' permission from your grader. It is acceptable to use calculators and computers on homework (indeed, some of it will require a computer) but calculators are not allowed during exams. You are encouraged to work together on homework, but writeups must be done individually.<br />
<br />
Many of the problems in this course will ask you to prove things. I expect proofs to be written in English sentences; the proofs in Stein's book are a good model for the level of verbosity I am looking for. <br />
<br />
'''Grading:''' The grade in Math 567 will be composed of 40% homework, 20% each of three midterms. The last midterm will be take-home, and will be due on the last day of class. ''There will be no final exam in Math 567''.<br />
<br />
'''Syllabus:''' <br />
(This may change as we see what pace works well for the course. All section numbers refer to Stein's book.)<br />
<br />
* Sep 3-10: Prime numbers, prime factorizations, Euclidean algorithm and GCD (1.1-1.2)<br />
* Sep 13-17: The integers mod n, Euler's theorem, the phi function (2.1-2.2)<br />
* Sep 20-24: Modular exponentiation, primality testing, and primitive roots (2.4-2.5)<br />
* Sep 27-Oct 1: Public-key cryptography and RSA (3.1-3.4)<br />
* Oct 4 - 8: Algebraic numbers <br />
* Oct 6: ''First midterm exam''<br />
* Oct 11-15: Quadratic reciprocity (4.1-4.4)<br />
* Oct 18-22: Finite and infinite continued fractions (5.1-5.3)<br />
* Oct 25-29: Continued fractions and diophantine approximation (5.4-5.5)<br />
* Nov 1-5: Diophantine equations I: Pell's equation and Lagrange's theorem<br />
* Nov 8-12: Elliptic curves (6.1-6.2)<br />
* Nov 10: ''Second midterm exam''<br />
* Nov 15-19: Applications of elliptic curves (6.3-6.4)<br />
* Nov 22-Dec 3: Diophantine equations II: Fermat, generalized Fermat, and probabilistic methods<br />
* Dec 6-15: advanced topic TBD: maybe a look at the Sato-Tate conjecture?<br />
<br />
<br />
<br />
'''Homework:'''<br />
Homework is due at the beginning of class on the specified Friday. Typing your homework is not a requirement, but if you don't already know LaTeX I highly recommend that you learn it and use it to typeset your homework. I will sometimes assign extra problems, which I will e-mail to the class list and include here.<br />
<br />
* Sep 13 (note this is Monday, not Friday!): 1.1, 1.3, 1.5, 1.7 (use SAGE), 1.8, 1.14.<br />
Problem A: Use SAGE to compute the number of x in [1..N] such that x^2 + 1 is prime, for N = 100, N = 1000, and N = 10000. Let f(N) be the number of such N. <br />
<br />
a) Can you formulate a conjecture about the relationship between f(N) and N/log N? <br />
<br />
b) What if x^2 + 1 is replaced with x^2 + 2? Can you explain why x^2 + 2 appears less likely to be prime? (Hint: consider x mod 3.)<br />
<br />
c) Prove that f(N) is at most (1/2)N+1. (Hint: consider x mod 2.) <br />
<br />
d) Give as good an upper bound as you can for f(N).<br />
<br />
Note that, despite the evident regularities you'll observe in this problem, we do not even know whether there are infinitely many primes of the form x^2 + 1! You would become very famous if you proved this.<br />
<br />
* Sep 17: 2.6 (the formulation of numerical evidence should be done by Sage if you've got Sage working, and by calculator if not; you can use an online tool like [http://primes.utm.edu/curios/includes/primetest.php this] to test whether a number is prime.) 2.8,2.9,2.11,2.12,2.14,2.19</div>Sjohnsonhttps://wiki.math.wisc.edu/index.php?title=Math_567_--_Elementary_Number_Theory&diff=833Math 567 -- Elementary Number Theory2010-09-08T19:54:15Z<p>Sjohnson: </p>
<hr />
<div>'''MATH 567<br />
<br />
Elementary Number Theory'''<br />
<br />
MWF 1:20-2:10, Van Vleck B119<br />
<br />
'''Professor:''' [http://www.math.wisc.edu/~ellenber/ Jordan Ellenberg] (ellenber@math.wisc.edu)<br />
''Office Hours:'' Weds, 2:30-3:30, Van Vleck 323. <br />
<br />
'''Grader:''' [http://www.math.wisc.edu/~sjohnson/ Silas Johnson] (snjohnson3@wisc.edu)<br />
''Office Hours:'' Thurs, 1:15-2:15, Van Vleck 522.<br />
<br />
Math 567 is a course in elementary number theory, aimed at undergraduates majoring in math or other quantitative disciplines. A general familiarity with abstract algebra at the level of Math 541 will be assumed, but students who haven't taken 541 are welcome to attend if they're willing to play a little catchup. We will be using William Stein's new (and cheap) textbook [http://www.amazon.com/Elementary-Number-Theory-Computational-Undergraduate/dp/0387855246 Elementary Number Theory: Primes, Congruences, and Secrets], which emphasizes computational approaches to the subject. If you don't need a physical copy of the book, [http://www.williamstein.org/ent/ it is available as a free legal .pdf.] We will be using the (free, public-domain) mathematical software [http://www.sagemath.org/ SAGE], developed largely by Stein, as an integral component of our coursework. There is a [http://www.sagemath.org/pdf/SageTutorial.pdf useful online tutorial.] You can download SAGE to your own computer or [http://www.sagenb.org use it online].<br />
<br />
Topics include some subset of, but are not limited to: Divisibility, the Euclidean algorithm and the GCD, linear Diophantine equations, prime numbers and uniqueness of factorization. Congruences, Chinese remainder theorem, Fermat's "little" theorem, Wilson's theorem, Euler's theorem and totient function, the RSA cryptosystem. Number-theoretic functions, multiplicative functions, Möbius inversion. Primitive roots and indices. Quadratic reciprocity and the Legendre symbol. Perfect numbers, Mersenne primes, Fermat primes. Pythagorean triples, Special cases of Fermat's "last" theorem. Fibonacci numbers. Continued fractions. Distribution of primes, discussion of prime number theorem. Primality testing and factoring algorithms.<br />
<br />
<br />
'''Course Policies:''' Homework will be due on Fridays. It can be turned in late only with ''advance'' permission from your grader. It is acceptable to use calculators and computers on homework (indeed, some of it will require a computer) but calculators are not allowed during exams. You are encouraged to work together on homework, but writeups must be done individually.<br />
<br />
Many of the problems in this course will ask you to prove things. I expect proofs to be written in English sentences; the proofs in Stein's book are a good model for the level of verbosity I am looking for. <br />
<br />
'''Grading:''' The grade in Math 567 will be composed of 40% homework, 20% each of three midterms. The last midterm will be take-home, and will be due on the last day of class. ''There will be no final exam in Math 567''.<br />
<br />
'''Syllabus:''' <br />
(This may change as we see what pace works well for the course. All section numbers refer to Stein's book.)<br />
<br />
* Sep 3-10: Prime numbers, prime factorizations, Euclidean algorithm and GCD (1.1-1.2)<br />
* Sep 13-17: The integers mod n, Euler's theorem, the phi function (2.1-2.2)<br />
* Sep 20-24: Modular exponentiation, primality testing, and primitive roots (2.4-2.5)<br />
* Sep 27-Oct 1: Public-key cryptography and RSA (3.1-3.4)<br />
* Oct 4 - 8: Algebraic numbers <br />
* Oct 6: ''First midterm exam''<br />
* Oct 11-15: Quadratic reciprocity (4.1-4.4)<br />
* Oct 18-22: Finite and infinite continued fractions (5.1-5.3)<br />
* Oct 25-29: Continued fractions and diophantine approximation (5.4-5.5)<br />
* Nov 1-5: Diophantine equations I: Pell's equation and Lagrange's theorem<br />
* Nov 8-12: Elliptic curves (6.1-6.2)<br />
* Nov 10: ''Second midterm exam''<br />
* Nov 15-19: Applications of elliptic curves (6.3-6.4)<br />
* Nov 22-Dec 3: Diophantine equations II: Fermat, generalized Fermat, and probabilistic methods<br />
* Dec 6-15: advanced topic TBD: maybe a look at the Sato-Tate conjecture?<br />
<br />
<br />
<br />
'''Homework:'''<br />
Homework is due at the beginning of class on the specified Friday. Typing your homework is not a requirement, but if you don't already know LaTeX I highly recommend that you learn it and use it to typeset your homework. I will sometimes assign extra problems, which I will e-mail to the class list and include here.<br />
<br />
* Sep 13 (note this is Monday, not Friday!): 1.1, 1.3, 1.5, 1.7 (use SAGE), 1.8, 1.14.<br />
Problem A: Use SAGE to compute the number of x in [1..N] such that x^2 + 1 is prime, for N = 100, N = 1000, and N = 10000. Let f(N) be the number of such N. <br />
<br />
a) Can you formulate a conjecture about the relationship between f(N) and N/log N? <br />
<br />
b) What if x^2 + 1 is replaced with x^2 + 2? Can you explain why x^2 + 2 appears less likely to be prime? (Hint: consider x mod 3.)<br />
<br />
c) Prove that f(N) is at most (1/2)N+1. (Hint: consider x mod 2.) <br />
<br />
d) Give as good an upper bound as you can for f(N).<br />
<br />
Note that, despite the evident regularities you'll observe in this problem, we do not even know whether there are infinitely many primes of the form x^2 + 1! You would become very famous if you proved this.</div>Sjohnsonhttps://wiki.math.wisc.edu/index.php?title=Math_567_--_Elementary_Number_Theory&diff=618Math 567 -- Elementary Number Theory2010-08-27T18:02:00Z<p>Sjohnson: </p>
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<div>'''MATH 567<br />
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Elementary Number Theory'''<br />
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MWF 1:20-2:10, Van Vleck B119<br />
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'''Professor:''' [http://www.math.wisc.edu/~ellenber/ Jordan Ellenberg] (ellenber@math.wisc.edu)<br />
''Office Hours:'' Weds, 2:30-3:30, Van Vleck 323. <br />
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'''Grader:''' [http://www.math.wisc.edu/~sjohnson/ Silas Johnson] (snjohnson3@wisc.edu)<br />
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Math 567 is a course in elementary number theory, aimed at undergraduates majoring in math or other quantitative disciplines. A general familiarity with abstract algebra at the level of Math 541 will be assumed, but students who haven't taken 541 are welcome to attend if they're willing to play a little catchup. We will be using William Stein's new (and cheap) textbook [http://www.amazon.com/Elementary-Number-Theory-Computational-Undergraduate/dp/0387855246 Elementary Number Theory: Primes, Congruences, and Secrets], which emphasizes computational approaches to the subject. If you don't need a physical copy of the book, [http://www.williamstein.org/ent/ it is available as a free legal .pdf.] We will be using the (free, public-domain) mathematical software [http://www.sagemath.org/ SAGE], developed largely by Stein, as an integral component of our coursework. There is a [http://www.sagemath.org/pdf/SageTutorial.pdf useful online tutorial.] You can download SAGE to your own computer or [http://www.sagenb.org use it online].<br />
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Topics include some subset of, but are not limited to: Divisibility, the Euclidean algorithm and the GCD, linear Diophantine equations, prime numbers and uniqueness of factorization. Congruences, Chinese remainder theorem, Fermat's "little" theorem, Wilson's theorem, Euler's theorem and totient function, the RSA cryptosystem. Number-theoretic functions, multiplicative functions, Möbius inversion. Primitive roots and indices. Quadratic reciprocity and the Legendre symbol. Perfect numbers, Mersenne primes, Fermat primes. Pythagorean triples, Special cases of Fermat's "last" theorem. Fibonacci numbers. Continued fractions. Distribution of primes, discussion of prime number theorem. Primality testing and factoring algorithms.<br />
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'''Course Policies:''' Homework will be due on Fridays. It can be turned in late only with ''advance'' permission from your grader. It is acceptable to use calculators and computers on homework (indeed, some of it will require a computer) but calculators are not allowed during exams. You are encouraged to work together on homework, but writeups must be done individually.<br />
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Many of the problems in this course will ask you to prove things. I expect proofs to be written in English sentences; the proofs in Stein's book are a good model for the level of verbosity I am looking for. <br />
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'''Grading:''' The grade in Math 567 will be composed of 40% homework, 20% each of three midterms. The last midterm will be take-home, and will be due on the last day of class. ''There will be no final exam in Math 567''.<br />
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'''Syllabus:''' <br />
(This may change as we see what pace works well for the course. All section numbers refer to Stein's book.)<br />
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* Sep 3-10: Prime numbers, prime factorizations, Euclidean algorithm and GCD (1.1-1.2)<br />
* Sep 13-17: The integers mod n, Euler's theorem, the phi function (2.1-2.2)<br />
* Sep 20-24: Modular exponentiation, primality testing, and primitive roots (2.4-2.5)<br />
* Sep 27-Oct 1: Public-key cryptography and RSA (3.1-3.4)<br />
* Oct 4 - 8: Algebraic numbers <br />
* Oct 6: ''First midterm exam''<br />
* Oct 11-15: Quadratic reciprocity (4.1-4.4)<br />
* Oct 18-22: Finite and infinite continued fractions (5.1-5.3)<br />
* Oct 25-29: Continued fractions and diophantine approximation (5.4-5.5)<br />
* Nov 1-5: Diophantine equations I: Pell's equation and Lagrange's theorem<br />
* Nov 8-12: Elliptic curves (6.1-6.2)<br />
* Nov 10: ''Second midterm exam''<br />
* Nov 15-19: Applications of elliptic curves (6.3-6.4)<br />
* Nov 22-Dec 3: Diophantine equations II: Fermat, generalized Fermat, and probabilistic methods<br />
* Dec 6-15: advanced topic TBD: maybe a look at the Sato-Tate conjecture?<br />
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'''Homework:'''<br />
Homework is due at the beginning of class on the specified Friday. Typing your homework is not a requirement, but if you don't already know LaTeX I highly recommend that you learn it and use it to typeset your homework. I will sometimes assign extra problems, which I will e-mail to the class list and include here.<br />
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* Sep 10: 1.1, 1.3, 1.5, 1.7 (use SAGE), 1.8, 1.14.<br />
Problem A: Use SAGE to compute the number of x in [1..N] such that x^2 + 1 is prime, for N = 100, N = 1000, and N = 10000. Let f(N) be the number of such N. <br />
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a) Can you formulate a conjecture about the relationship between f(N) and N/log N? <br />
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b) What if x^2 + 1 is replaced with x^2 + 2? Can you explain why x^2 + 2 appears less likely to be prime? (Hint: consider x mod 3.)<br />
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c) Prove that f(N) is at most (1/2)N+1. (Hint: consider x mod 2.) <br />
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d) Give as good an upper bound as you can for f(N).<br />
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Note that, despite the evident regularities you'll observe in this problem, we do not even know whether there are infinitely many primes of the form x^2 + 1! You would become very famous if you proved this.</div>Sjohnson