Announcements and handouts will be available on this page. Since announcements may be posted at any time, you should check this page frequently. However, I will try to make sure that important announcements and handouts are mentioned in the RSS feed. Make sure you hard refresh this page to ensure that you are not reading a cached version. Handouts will be PDF files, but are available to students in other formats upon request.
April's comments.25 April.
The statistics sheet for the final exam is now available.
Thanks to everyone for making this such a great class. I had a lot of fun this semester, and I hope you did, too.
As mentioned in Tuesday's review session, students must also be able to prove the second part of the Chinese remainder theorem (the part concerned with how nearly unique is a solution to a system of congruences). The review sheet has been updated to reflect this.
Also, Lar Biederman pointed out a few typoes in the homework solution sheets. A few of the homework sheets incorrectly said that they were for Homework 8. Also, some of the Euclidean algorithm tables on the solution set for Homework 10 had spurious entries in the second row of the Q column. These typoes have been corrected.
15 April (8 PM).
The review sheet for the final exam is now available. The list of practice questions has been updated slightly.
15 April (5 PM).
I have posted a list of practice questions for the final exam. These are mostly focussed on new material (since the second exam), and, by request, on questions about abstract functions. There are a few (optional) difficult questions, marked with a *.
15 April (3 PM).
In the rush of material before the second exam, I forgot to mention this in the comments, but thanks to Lar Biederman for pointing out a typo in the statement of Problem 126.96.36.199(a) on Homework 5. We were supposed to find a function with range R, not range Z. This has been corrected.
In Tuesday's class, we did some more examples of how to identify quadratic residues modulo odd prime powers, and how to find square roots of those numbers which are quadratic residues, but are not divisible by the corresponding prime. In Thursday's class, we discussed the fact that the algorithm for finding square roots leads us (somewhat) naturally to the concept of the p-adic numbers Qp, which we defined briefly (but which will not appear on the final). We also walked through finding a method of converting a date into a day of the week. The method we came up with is essentially the one of Section 6.1.2, except that we fixed a typo (the formula in the book should have a term of y in it, to account for the fact that moving forward one year usually moves forward one day of the week).
15 April (12:01 AM).
Solution sets are now available for homeworks 9, 10, and 11. As before, they have been only lightly proofread, so be sure to mention to me any errors that you find.
Office hours during the week of the final will be held 3--4 PM on Tuesday, 2:30--3:30 PM on Wednesday, and 2:30--3:45 PM on Thursday. Tuesday's office hours are at the normal time, but Thursday's office hours start and end earlier than usual. The exam itself is 4--6 PM on Thursday, in 1360 EH.
If you want your exam mailed to you after I grade it, please give me a self-addressed stamped envelope by the day of the final. You can also pick up your exam by appointment some time in the year after the final.
If you would like your grade e-mailed to you when I have assigned it, please give me a signed request with your e-mail address on it. I cannot accept such requests by e-mail.
The following people have expressed an interest in participating in study groups:
Lar Biederman (firstname.lastname@example.org)
Daniel Chun (DanielC@Review.com)
Mike Cook (email@example.com)
Sarah Dee (firstname.lastname@example.org)
Megan King (email@example.com)
Lisa Kurtz (firstname.lastname@example.org)
There was a slight difference between the homework that was assigned in class, and the homework that appears on the web page. If you have already done the homework assigned in class, that's OK, since it is strictly stronger; but, if not, then you should just do the homework as it appears on the web page. In particular, the problem about square roots modulo odd prime is just an 'if' question, not an 'if and only if' one; and the problem about finding a square root of 2 modulo 17, if there is one, is not part of the homework (though it is good practice).
I have been asked to inform students of a possible job opportunity. A recruiter from the Los Angeles Unified School District will be interviewing certified secondary school teachers in Detroit on May 18 and 19, 2007. Please see the file DETROIT FLYER 2007.doc linked above for more details.
For the week of 2--6 April, I will be out of town. Alexei Kolesnikov has agreed to make his office hours (Monday 1--3 PM and Friday 1--2 PM, in the math lab) available for students in this class. I will also answer e-mail questions as promptly as possible.
The following three problems will be assigned this week.
Suppose that m is an odd prime, and that x and y are integers such that x2 ≡ y2 (mod m). Show that x ≡ ±y (mod m). Show, by example, that this can fail if m is only required to be an odd integer.
Suppose that p is an odd prime, a is not congruent to 0 modulo p, and there is an integer x such that x2 ≡ a (mod p). Show that a(p - 1)/2 ≡ 1 (mod p). (Try using Fermat's little theorem.)
What are the least non-negative quadratic residues modulo 25 which are not congruent to 0 modulo 5? Find square roots of 14 modulo 5, 25, and 125.