University of Massachusetts, Amherst
Math 471

Theory of Numbers
Fall 2012

Click here to go the Homework Page

Course News:
Monitor this space for important course news, such as...

Sep. 20, 2012: MIDTERM DATES ARE THURSDAY OCT 11 [Note change!] & THURSDAY NOV 8. The exams are in-class.

Sample Exam 1 has been posted; it is also HW5, due on Oct. 11 prior to start of exam.

Note that we do NOT have class on Tuesday Oct. 9, because on that Tuesday, we pretend Tuesdays are Mondays.

Oct 9: We have a room for our Review Session! Lederle 219, Wed Oct 10, 4:00-5:00 pm.

December 10: We have a room for the Final Exam Review Session! Lederle 219, Tues. December 11, 3:00-5:00.

And don't forget the fabulous Final Exam itself will take place on Wednesday December 12, bright and early 8-10 am, in Lederle A201 (Lowrise).

Meeting times: Tuesday and Thursday, 1:00-2:15, in Lederle 202.

Instructor: Dr. Farshid Hajir
Office: Lederle 1623C
Phone: 545-6025
Email: hajir atsymbol

Office Hours: Current office hours are Tuesday 11-12 , and Wednesday 2-3, subject to change. You are always welcome to set up an appointment by sending me an e-mail or calling me on the phone.

Text: A Friendly Introduction to Number Theory by Joe Silverman, Pearson, Edition: 4th, Year Published: 2012, Price: 105.00 USD. Yes, it's somewhat pricey, but the quality of the book is very high. I will assign a certain amount of reading and homework from this text. If you want a cheap standard book on number theory, you can considering buying Fundamentals of Number Theory by William J. LeVeque, Dover Publications, ISBN 0-486-68906-9 (paperback) $9.95.
Finally, It is out of print, but if you can find a copy, the book by Dan Flath "Introduction to Number Theory" (Wiley, 1988) ISBN 047160836X is a helpful resource. Additionally, I may post my own course notes to the course website. [If you are assigning probabilities to the previous statement, you may also consider that cows could one day possibly fly.]

Moodle Bulletin Board: A virtual M471 discussion may be set up via the Campus Moodle resource. Stay tuned for details. Please use this service responsibly. (Assuming I actually set it up, I will monitor it semi-regularly, but if you want to direct a question specifically at me, the best way to reach me is during office hours or by e-mail.

Homework: Homework will be posted on The Homework Page and collected every Tuesday at the beginning of lecture. Late homework will not be accepted and the lowest homework grade will be dropped. Be sure to read and follow the homework rules.

Attendance: Attendance is required during lectures. I consider attendance AND participation important ingredients for your success in the course. Frequent absences will be reflected in your grade.

Quizzes: I might give an occasional 10-15 minute quiz in class. Each of these will count as one homework assignment. There will be no make-up quizzes.

Exams: There will be two midterms and one final exam. The midterms will take place during regular class time on Thursday October 4 and Thursday November 8. The final will be on a date to be determined during the regular final-exam period. Make-up Exam Policy: If you have a legitimate (e.g. multiple exams at the same exact time, medical problems, emergency absences, religious observances) scheduling conflict with any of the exams, it is your responsibility to notify me of the conflict as soon as you become aware of it. The final decision regarding allowing a make-up exam is mine, subject to University regulations, of course. Please note that previously arranged travel plans are not a valid reason to be given a make-up exam.

Computers: Number theory is an inherently computational subject and we will undoubtedly have some use for machine calculations over the course of the semester. When the time comes, you will probably be able to get away with Maple or Mathematica if you're used to those packages. In fact, however, there is a far better option out there: Pari/GP. Pari/GP has two main advantages: 1) it is specifically designed for use in number theory; and 2) it is available at a very reasonable price: free. We'll deal with this when the time comes, but if you're curious, everything you ever wanted to know about Pari/GP can be learned at The PARI homepage.

Extra Credit: Some extra credit problems will occasionally be included in the homework assignments, or given during class.  The number of points for each problem will vary, as will the difficulty of the problem. The student with the most points at the end of the semester wins a fabulous prize. You may hand in Extra Credit solutions at any time throughout the term, until the last class meeting.

   homework, quizzes, participation - 30%
   2 midterms - 20% each
   Final exam - 30%

Grading Scale


≥ 93%


≥ 90% and < 90%


≥ 86% and < 90%


≥ 82% and < 86%


≥ 78% and < 82%


≥74% and < 78%


≥ 70% and < 74%


≥65% and < 70%


≥ 60% and < 65%


< 60%

Course topics:
Motivating questions: Pythaogrean triples, Fermat's last theorem, cryptography.
Divisibility: Divisibility, greatest common divisor, Euclid's algorithm, linear equations, Unique Factorization Primes, fundamental theorem of arithmetic.
Congruences: Congruences, Fermat's little theorem, Euler's theorem, Chinese remainder theorem, powers, successive squaring.
Prime numbers: Counting primes, Mersenne primes, perfect numbers.
Cryptography: RSA cryptography, Carmichael numbers, primality tests.
Primitive roots: Primitive roots, discrete logarithms.
Quadratic reciprocity: Quadratic residues, quadratic reciprocity.
Gaussian integers: Sums of squares, Gaussian integers, arithmetic of Gaussian integers.
Diophantine approximation: Pell's equation, diophantine approximation.
Additional topics as time permits.