Mathematics 115
Fall, 1999
3 Evans Hall, TuTh 12:40-2

Professor Ken Ribet

885 Evans Hall

Office hours: M 10-11, W 11-12, Th 10:30-11:30
Office telephone: 642 0648
Fax number: 642 8204
Secretary: 642 5026
email: ribet@math.berkeley.edu

Textbook

Some software that may be run on a PC under DOS is available by ftp to ftp.math.lsa.umich.edu. The README explains how to run the binary executable that you can download from the server. For more extensive documentation, pick up the files clint0.pdf, clint1.pdf, clint2.pdf, clint3.pdf, and clint4.pdf. Our book's author Hugh Montgomery wrote to me as follows:

In this set of programs is one that performs powering congruentially for integers up to 10¹⁸. I wrote the programs in Turbo Pascal almost 10 years ago. Recently I learned that the Borland Turbo Pascal compiler has a bug in it that causes programs compiled with it to crash on PCs running at clock speeds of (roughly) 300MHz or higher. I have installed a patch in my compiler that fixes this, but I haven't gotten around to recompiling the programs that are there.

While I'm at it, I might refer you to Prime Form, a primality-testing program that is available "for all 32-bit Windows operating systems and has been tested under Windows 95, 98, NT3.51, NT4.0, and Windows 2000."

According to the General Catalog, this course treats ``Divisibility, congruences, numerical functions, theory of primes. Topics selected: Diophantine analysis, continued fractions, partitions, quadratic fields, asymptotic distributions, additive problems.''

Examinations

• First Midterm: September 23, in class (questions and answers). Here's how the scores were distributed:

  
  
                                     X                          
                         X       X   X                          
                 X       X X X X X   X X   X X       X       X  
       X X   X   X   X X X X X X X X X X X X X X X   X X     X  
   |-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-| 
   5        1 0       1 5       2 0       2 5       3 0       3 5
  
  
  Median = 20. 5
  Average = 20.8
  Maximum possible score = 35
  

• Second Midterm, October 28, in class (questions and answers). Here's how the scores were distributed:

  
  
                                       X           X            
                                       X X         X X
                             X X X X X X X     X   X X
           X X       X   X   X X X X X X X X X X X X X X     X X
   |-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-| 
   5        1 0       1 5       2 0       2 5       3 0       3 5
  
  
  Median = 23. 5
  Average = 23.67
  Maximum possible score = 35
  

• Final Exam, Tuesday December 14, 12:30-3:30, in 2 Evans (questions and answers). Here's how the scores were distributed:

  
  
  
                                     x     x                             x       
  <-x (score=18)                     x   x x   x   x     x   x         x x     x 
   x                 x           x x x x x x   x   x   x x x x x   x x x x     x
  -|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|
  2 0       2 5       3 0       3 5       4 0       4 5       5 0       5 5
  
  
  
  Median = 43
  Average = 43.74
  Maximum possible score = 60
  

Homework

For numerical problems, the grader encourages you to use computer software as you see fit. Be sure, however, to include printouts that explain what you did.

1. Assignment due September 2:
   • § 1.2: 1 (b, d), 3 (b,c), 4b, 9, 13, 14.
2. Assignment due September 9:
   • § 1.2: 24, 25, 46
   • § 1.3: 5, 7, 8, 11, 26, 27, 42, 43, 44
3. Assignment due September 16:
   • Suppose that a and b are relatively prime integers for which the fraction a/b is the nth power of a rational number (n a positive integer). Suppose that b is positive. Show that a and b are each perfect nth powers of integers.
   • § 1.2: 47, 49, 50
   • § 1.3: 51
   • § 2.1: 5, 6, 9, 12, 13, 14, 17, 23, 27, 28, 38
4. Assignment due Tuesday, September 28:
   • § 2.1: 40, 41, 43, 44, 47, 48
   • § 2.2: 4, 5efg, 6ab, 7, 8, 9
5. Assignment due Thursday, October 7:
   • § 2.2: 10, 14, 15
   • § 2.3: 1, 3, 4, 7, 9, 15, 18

6. Assignment due Thursday, October 14:
   • In connection with the RSA algorithm (§ 2.5), work out numerically what happens if m=391, the exponent k (called e in my lecture) is 3, and the message to be sent is "123". Calculate the encrypted message (123)^k mod m and then decrypt this message.
   • § 2.5: 2, 3, 4
   • § 2.3: 19, 28, 29
   • § 2.4: 2, 3 (all parts), 4, 5, 6
   Author Hugh Montgomery suggests that Problem 3 in § 2.4 be worked out using the programs that he wrote for IBM compatibles some years ago. For documentation, see the parent directory.

7. Assignment due Tuesday, October 26:
   • Study for the second midterm (October 28, in class)
   • § 2.4: 7, 9, 10, 13, 14a
   • § 2.6: 2, 3, 4
8. Assignment due Tuesday, November 9:
   • § 2.7: 6, 7, 8, 10
   • § 2.8: 2, 3, 4, 5, 8ab, 9, 16, 17, 18
9. Assignment due Tuesday, November 16:
   • § 3.1: 8 ace, 14, 15, 19
   • § 3.2: 2, 4ace, 6, 7, 10, 11, 13
10. Assignment due Thursday, December 2:
    • § 3.2: 15
    • § 3.3: 4, 7, 8, 9
    • § 7.1: 1, 3
    • § 7.2: 1
    • § 7.3: 2
    • § 7.4: 1
    • § 7.5: 3, 4
    • § 7.6: 1, 2
    • § 7.7: 1, 2, 3
    • § 7.8: 2, 3, 4, 5, 6
    [The problems in § 7.8 should be done by you in private after the lecture on December 2. They do not need to be submitted.] This is a long assignment, so please get started ASAP!

E-mail Messages

• September 11, 1999

Grading

Miscellaneous links related to number theory and/or this course

• TeachMath at Berkeley
• Least primitive root of prime numbers
• Teach for America (February 13, 1998)
• Large prime discovered by 19-year-old sophomore (February 3, 1998)
• Fermat's Last Theorem T-shirt (we will put in a group order if there's interest)
• Number Theory Web
• Life by the Numbers

Kenneth A. Ribet , Math Department 3840, Berkeley CA 94720-3840

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