Math 25
Number Theory
Last updated May 31, 2008 12:24:21 EDT

Announcements:

• The Second Midterm has been graded. The average score was 62.3 out of 80. Solutions to the exam are available in the Examinations section of the General Info page.
• The take-home final exam will be distributed in class on Monday, November 29. It is due at the end of the scheduled final exam period: Saturday, December 4, 11 am.

### Homework Assigments

Week of September 20 - 24, 2004
All problems assigned this week are due on Wednesday, 9/29.
Solutions to this assignment are available here.
 Monday: Study: No class Do: Whatever you see fit... Wednesday: Study: Sections 1.2 and 1.4 Do: pp 22 - 23: 3, 15, 24 pp 34 - 35: 6, 8, 17 Friday: Study: Section 3.1 Do: pp 76 - 78: 2, 11, 13, 21, 24, 29 Plus: Show that for any real number x the quantity [2x]- 2[x] is either 0 or 1.

Week of September 27 - October 1, 2004
(Due Wednesday, October 6)
Solutions to this assignment are available here.
 Monday: Study:Section 3.2 Do: pp 84 - 85: 5, 8, 9, 20, 22, 31 Wednesday: Study:Sections 3.3, 3.4 Do: pp 104 - 109: 8, 13, 16 (only the base 10 part), 34, 43 (part b does not need part a), 69 Friday: Study: Sections 3.5, 3.6, 13.1 (only through Lemma 13.3) Do: pp 117 - 119: 4 (you may assume all the numbers given are prime or products of two primes), 24 pp 123 - 125: 2, 4, 13 p 486: 2

Week of October 4 - 8, 2004
(Due Wednesday, October 13)
Solutions to this assignment are available here.
 Monday: Study: Sections 13.1, 13.2, and the beginning of 4.1 Do: pp 486 - 487: 5, 12 pp 494 - 495: 4 (just give it a shot; it might be rather hard), 10 pp 135 - 138: 3, 8, 10, 17 Wednesday: Study: Section 4.1 Do: pp 135 - 138: 3, 5, 7, 8, 17, 28c Friday: Study: Section 4.2 Do: pp 135 - 138: 16, 22 pp 141 - 143: 2, 6, 16, 18 pp 149 - 151: 14 (Since we don't know CRT yet, solve this using Dirichlet's theorem on primes in progressions.

Week of October 11 - 15, 2004
(Due Wednesday, October 20)
Solutions to this assignment are available here.
 Monday: Study: Sections 4.2 and 4.3 Do: pp 141 - 143: 8,10 pp 149 - 152: 10, 12, 18, 22, 34 Wednesday: Study: Section 4.4 Do: pp 149 - 152: 14 (do this problem again, but using CRT this time), 30 pp 158 - 159: 2, 8 Friday: Study: Section 4.4 Do: pp 158 - 159: 6 Plus: These additional problems.

Week of October 18 - 22, 2004
(Due Wednesday, October 27)
 Monday: Study: Sections 5.1 and 5.2 Do: pp 34 - 37: 31. Show also that the result of this exercise implies that [C/4 + Y/400] = [C/4] (notation from lecture; see also p 181). pp 177 - 179: 2, 4, 17, 19, 22 Wednesday: Study: Section 5.5 Do: p 183: 3, 7 pp 194 - 196: 2, 6, 12, 14, 20 (optional) Friday: Study: Section 6.1 Do: pp 202 - 204: 4, 16, 20, 24 Plus: These problems on the formula for the nth prime number

Week of October 25 - 29, 2004
(Due Wednesday, November 3)
 Monday: Study: Sections 4.6 and 6.1 Do: p 172: 1e, 1f pp 202 - 204: 26, 27, 28, 32, 34 (part (a) of this problem is *false* as stated. The hypothesis should be (-1)^r * r! congruent to 1 mod p. The conclusion is wrong, too. What should the conclusion actually be?) Wednesday: Study: Section 6.2 Do: pp 213 - 214: 2, 8, 16, 18, 20 Friday: Study: Section 6.3 Do: pp 213 - 214: 13, 14 pp 218 - 219: 4, 6, 10, 12

Week of November 1 - 5, 2004
(Due Wednesday, November 10)
 Monday: Study: Sections 8.1, 8.2 Do: pp 267: 6, 10, 12, 16 pp 278 - 281: 2, 6, 14 Wednesday: Study: Section 8.2 Do: Something fun. No homework assigned today. Friday: Study: Notes Do: These problems on linear systems of congruences and cryptanalysis. You will need to use Maple (the software, not the syrup) to do these problems. Dartmouth students can download it for free here. You will also need this Maple file of Cryptographic tools. To use it, open it and press enter repeatedly until you get to a blank command prompt. (Slightly) more detailed instructions are available here. Troubleshooting: If you have trouble using the Maple file, make sure its name ends with ".mw". Some web browsers change the extension to ".xml" when the file is downloaded. If this doesn't help, try downloading this text file containing all the Maple code. To use it, copy each block of code into Maple, pressing enter after each block. If you have any other problems using Maple, let me know ASAP.

Week of November 8 - 12, 2004
(Due Wednesday, November 17)
 Monday: Study: Section 8.2 Do: pp 278 - 281: 8, 28 (you should use the Maple procedures available above for these), 10, 16 Wednesday: Study: Sections 8.3 and 8.4 Do: p 284: 4,6 pp 290 - 291: 2, 8 Plus: These additional problems. Friday: Study: The Beginning of Section 7.1 Do: pp 227 - 231: 40, 42, 44 Also: Prove the converse of Exercise 44: If f(1) = 1 and f has the factorization property of the exercise, then f is completely multiplicative.

Week of November 15 - 19, 2003
(Due Monday, November 29)
 Monday: Study: The rest of Section 7.1, Section 7.2 Do: pp 227 - 231: 6, 8, 16, 28, 30 Wednesday: Study: Notes, Sections 7.2 and 7.4 Do: pp 235 - 236: 4, 12 pp 256 - 257: 10, 17, 18, 20, 26 (use the method of Euler products introduced in class for 18, 20 and 26), Friday: Study: Notes, beginning of Section 2.3 Do: These problems on square-free integers.

Week of November 22 - 26, 2004
(Due Wednesday, December 1)