Math 453 C3M Home Page

This will be the home page for Math 453 D13 "Elementary Theory of Numbers". This class meets for the Fall 2019 semester at MWF 11:00-11:50 in 343 Altgeld. Office hours to be announced.
The mathematical links page will be here. Suggest a new one!

Many students are concerned about the political issues of the day. If you want to contact your representatives, here are their numbers:
US Sen. Tammy Duckworth (D-IL) (202)-224-2859
US Sen. Richard J. Durbin (D-IL) (202)-224-2152
US Rep. Rodney Davis (R-IL 13) (202)-225-2371
IL Sen. Scott Bennett (D-IL 52) (217)-335-5252
IL Rep. Carol Ammons (D-IL 103) (217)-531-1660

Homework 3 comments and corrections

  • Yes, there's a typo in #7(ii). That should be r, s > 2. The problem is false in case, e.g. r=2, s = 3. My apologies.

    W 9/18 -- The first hour test is scheduled for Friday, October 4. Wilson's Theorem and Fermat's Little Theorem were the main topics; there was a review of what we've been doing with solving ax = b mod m and the Chinese Remainder Theorem, and a bio sketch of former UIUC prof (1915-1947) Robert Carmichael (1879-1967), who discovered Carmichael numbers.

    M 9/16 -- The filled-in worksheet was distributed. This contained a solution to another CRT problem. In class, I went through the proof of the CRT in excruciating detail, solved Sun Zi's problem in two ways, and got close to proving Wilson's Theorem, and sections 2.4 and 2.5. Don't be shy about emailing me questions.

    F 9/13 -- The Chinese Remainder Theorem. There was a related Worksheet which occupied the end of the hour. The Second Homework Solutions were distributed. (#4 is corrected on this!) The Third Homework was distributed, due F 9/13. On Monday, continue through the Chinese Remainder Theorem and its implications.

    W 9/11 -- Lots of examples of congruences and the solution of congruential equations, with data from the class. We are about to start the Chinese Remainder Theorem. Here is a biography of Sun Zi.

    M 9/9 -- Phone and email list distributed (no links), also a first handout on v_p(n) and its applications. We started in on congruences from sections 2.1 and 2.2 and are in the middle of the proof of Theorem 2.6, having already done so when gcd(a,m) = 1.

    F 9/6 -- Phone and email list circulated for the second and last time. GCD's and the graphical interpretation of divisibility of numbers of the form 2^a3^b. There was a related Worksheet which occupied the end of the hour. The First Homework Solutions were distributed. The Second Homework was distributed, due F 9/13. On Monday, we start Chapter 2 of the book.

    HW 1 Comments

  • If you want to see more examples of the Euclidean algorithm, go to the examples here. You can ignore the coding.

    W 9/4 -- We covered 1.5 up to p.28, plus problem 42 (p.21) with proof. I didn't get to the lcm, but I did define v_p(n), which is the exponent of the power p in the prime factorization of n. Prop. 1.17 and Thm. 1.19 are much "cleaner" that way. Friday will see the lcm, more on v_p(n) and a shortish worksheet at the end of the hour. Homework 1 will be due, and, I hope, Homework 1 solutions and Homework 2 questions will be distributed.

    F 8/30 -- Office hours will be Tues (5-5:30) in 443 AH and, most weeks, Mon (4-4:30) in 447 AH, except when the Senate is in session. The gcd and the Euclidean algorithm, with examples. I phrased Prop. 1.11 as aZ + bZ = gZ, where g = gcd(a,b). Next stop: the Fundamental Theorem of Arithmetic about unique factorization of positive integers into products of primes. The First Homework was distributed, due F 9/6.

    W 8/28 -- Talked about tentative test times. More explanation of f(x) = [x] (the floor function), and a lot of discussion about primes, both historical and otherwise. Most of it is in the book. The Prime Page is in the Fall 19 Links page cited above. Homework 1 will be out tomorrow and due 9/6. Also, office hours will be discussed.

    M 8/26 -- Introductions. Handouts were: Course Organization, How to Solve It guide, Class questionnaire. and information about the links page (see above.) We covered basically pp.1-10 of the text. Let me know if you have any questions. Notations not in the book: for a set A, x + A = {x+a:a in A}, for sets A, B, A + B = {a+b:a in A, b in B} (I forgot to give that one), and for n > 0, D(n) = {d : d | n}. We agreed to have homeworks due on Fridays. Let me know if I forgot anything. To answer an email question, the sums in A+B are taken over all elements independently, so if A = {1,2} and B = {10,20,30}, then A+B={11,12,21,22,31,32}.