Math 114: Explorations in Number Theory

Fall 2001

Schedule

Homework is due at each class meeting, unless stated otherwise.

Monday, Sept 10, 2001

• Homework: Read "In Code" pages 3-11 and think about puzzle. Work on Jailor puzzle. Teach someone the horse puzzle and Gauss' trick for adding up sequential numbers. Bring in signed note from your student saying what they thought of the tricks.

Wednesday, Sept 12, 2001

• Homework: Read "In Code" pages 12-top of 22 and think about the nine puzzles in this section. Write up on a piece of paper the solutions to the three puzzles: Rabbit puzzle, 100 meter dash puzzle, and Insurance puzzle. Teach someone one of the puzzles. Work on Jailor puzzle.

Monday, Sept 17, 2001

• Homework: Read "In Code" pages 22-31. Then complete the following problems: 1) Factor the numbers 2002, 840, 5040, 45670 into their primes. 2) Find the greatest common divisor of 2002 and 840.

Wednesday, Sept 19, 2001

• Homework: Read "In Code" pages 32-46 (Chapters 3 and 4). Then consider the numbers written out in 10 columns upon division by 10. So that you have a column of numbers of the form 10K, 10K+1, 10K+2,..., and 10K+9. If you square the numbers in each column which column do the squares end up in. After you work this out then answer the following questions: Which columns would you expect to see squares in and which columns would you expect to see no squares in? Would you expect to see more squares in some columns than in others?

Monday, Sept 24, 2001

• Homework: Read "In Code" pages 47-59. Then write out the addition and multiplication tables for arithmetic modulo 7.

Wednesday, Sept 26, 2001

• Homework: Read "In Code" pages 60-70. Then teach someone why there are an infinite number of primes. Teach them both proofs and bring back a signed note from them about what they learned and which proof they liked best.

Monday, Oct. 1, 2001

• Homework: Read "In Code" pages 71-87. Then write out the Cayley tables for the symmetries of the shapes we discussed in class.

Wednesday, Oct. 3, 2001

• Homework: Read "In Code" pages 87-98. 1. Write a short message using the Caesar Cipher. 2. Go on the internet and find the website "The great Mersenne Prime search". Find the largest prime known from that site. 3. Go through the handout on logos and designate each figure as C_n or D_n according to its symmetry group.

Friday, Oct. 5, 2001

• Homework: Read "In Code" pages 98-112. Find the additive inverses, and the multiplicative inverses (when they exist) for all the elements in Z_7 and Z_{11}.

Wednesday, Oct. 10, 2001

• Homework: Complete the sample exam as review for next Monday's exam.

Friday, Oct. 12, 2001

• Homework: Find the additive inverses, and the multiplicative inverses (when they exist) for all the elements in Z_{23}.

Monday, Oct 15, 2001: Review for the exam. TEST #1

Friday, Oct. 19, 2001

• Homework: Read and think about "In Code" pages 113-124. Then write out the multiplicative Cayley tables for Z_{12}*, Z_9*, and Z_{14}*. Compute the following congruences:

  456+789 =_____ modulo 12

  49^{12} =______modulo 10

  777^{777} = ______ modulo 10

Wednesday, Oct. 24, 2001

• Homework: Read and think about "In Code" pages 125-134. Find multiplicative orders of all the elements in Z_12*, Z_9*, Z_14*, Z_11*, Z_10*, and Z_8*. Compute the following congruences:

  149+354 =_____ modulo 14

  149^{483} =______modulo 10

  149^{483} =______modulo 5

  149^{483} =______modulo 6

  149^{483} =______modulo 11

• Homework: Read and think about "In Code" pages 134-148. Compute the following congruences:

  32^{439} =_____ modulo 7

  33^{439} =______modulo 13

  2^{439} =______modulo 14

• Homework: Read "In Code" Appendix C pp.305-310. Do Worksheet on the Euclidean Algorithm.

Monday, Nov. 5, 2001

• Homework: Read "In Code" Appendix C pp. 310-314. Do Worksheet on finding multiplicative inverses by working back up through the Euclidean Algorithm. Find solutions to 45x+23y=1, 126x+31y=1,58x+33y=1,165x+133y=1 and interpret the answers as multiplicative inverses in the correct Z_n*.

Wednesday, Nov. 7, 2001

• Homework: Read "In Code" Pages 149-164. Write out a definition for a probable prime and a definition for a pseudoprime. Show that 25 is a probable prime base 7. Compute:

  2^{35} =_____ modulo 11

  50^{40} =______modulo 19

  3^{45} =______modulo 13

  60^{70} =_____ modulo 19

  9^{42} =______modulo 43

  Find the multiplicative inverse of 111 in Z_676*.

Friday, Nov. 9, 2001

• Homework: Go over Fermat's and Wilson's Theorem. Compute: 100!=____mod 101; 9!=______mod 10; 20!=______mod 21; 8!=_____mod 9.

Monday, Nov. 12, 2001

• Homework: Read Chapter 9 in "In Code". Do practice problem worksheet:

  1. Find last digit of 2^50, and 1989^1989.

  2. Find remainder of 2^100 modulo 3

  3. Use Fermat's Little Theorem to find 874^339=____modulo 43

  4. Use the Euclidean Algorithm and working back up to find the solution to 11x+3y=1. Use the solutions to find the multiplicative inverse of 3 modulo 11.

  5. Find the elements in Z_14* and find all multiplicative inverses, all multiplicative orders, and all mutiplicative generators in Z_14*.

Monday, Nov. 19, 2001

• Homework: Read Chapter 9 in "In Code". Finish up back homework and start work on sample Exam II to prepare for exam after Vacation.

Thanksgiving Break

Friday, Nov. 26, 2001: TEST #2

Monday, Dec. 3, 2001

• Homework: Read Chapter 9 and Appendix D pp.315-319 in "In Code". Compute the following:

  1. phi(35), phi(77),phi(91), phi(9), phi(25), phi(27), phi(125),phi(49), phi(100),phi(36).

  2. Use Euler's theorem to compute:

  2^8 =______modulo 15

  2^9 =______modulo 15

  2^17 =_____ modulo 15

  3^60 =______modulo 77

  3^61 =_____ modulo 77

  3^121 =______modulo 77

Wednesday, Dec. 12, 2001: LAST DAY OF CLASS

Friday, Dec. 21, 2001: EXAMS NEED TO BE TURNED IN BY NOON