Math 255: Elementary Number Theory - Spring 2022

Work due this week:

• You can turn in a metacognition essay on Gradescope
  (due Wed May 11 at 11:59pm)

No new homework or material anymore.

No more projects.

Work due this week:

• Read this handout on continued fractions, then answer Reading quiz 12 on Gradescope
  (due Wed Apr 27 at 9:40am)
• There will not be in-class activities this week. (Please see announcement on Teams.)
• You can turn in a metacognition essay on Gradescope
  (due Sun May 1 at 11:59pm)

Homework (due May 11):

• Continued fractions by hand (15 points)
• Continued fractions with a calculator (15 points)

No project assigned this week, but if you think of any topic related to continued fractions that you would like to explore, please contact me and we can set it up as an assignment.

Work due this week:

• Read Section 4 of this handout on quadratic congruences, then answer Reading quiz 11 on Gradescope
  (due Mon Apr 18 at 9:40am)
• There will not be in-class activities this week. (Please see announcement on Teams.)
• You can turn in a metacognition essay on Gradescope
  (due Sun Apr 24 at 11:59pm)

Homework (due May 11):

• x^2 = a mod n (15 points)
• Quadratic congruences proofs* (10 points)

No project assigned this week, but if you think of any topic related to quadratic congruences that you would like to explore, please contact me and we can set it up as an assignment.

Work due this week:

• Read Sections 1, 2 and 3 of this handout on quadratic congruences, then answer Reading quiz 10 on Gradescope
  (due Mon Apr 11 at 9:40am)
• Don't forget to turn in your in-class activities, in person or on Gradescope!
  (due Sun Apr 17 at 11:59pm)
• You can turn in a metacognition essay on Gradescope
  (due Sun Apr 17 at 11:59pm)

Homework (due May 11):

• x^2 = a mod p^k (10 points)
• x^2 = a mod 2^k (10 points)

No project assigned this week, but if you think of any topic related to quadratic congruences that you would like to explore, please contact me and we can set it up as an assignment.

Work due this week:

• Read Section 11, then answer Reading quiz 9 on Gradescope
  (due Wed Apr 6 at 9:40am)
• Don't forget to turn in your in-class activities, in person or on Gradescope!
  (due Sun Apr 10 at 11:59pm)
• You can turn in a metacognition essay on Gradescope
  (due Sun Apr 10 at 11:59pm)

Homework (due May 11):

• The Legendre symbol (15 points)

As a projet this week, please read the proof of quadratic reciprocity in the book, and write down your understanding of it. More details can be found in this assignment sheet: The proof of quadratic reciprocity.

Work due this week:

• Read Section 10, then answer Reading quiz 8 on Gradescope
  (due Mon Mar 28 at 9:40am)
• Don't forget to turn in your in-class activities, in person or on Gradescope!
  (due Sun Apr 3 at 11:59pm)
• You can turn in a metacognition essay on Gradescope
  (due Sun Apr 3 at 11:59pm)

Homework (due May 11):

• The order of a modulo n* (10 points)
• Solving equations using primitive roots (10 points)
• Some cryptography (10 points)

No projects assigned this week, but check back later, I might think of something!

Work due this week:

• Read Section 9, then answer Reading quiz 7 on Gradescope
  (due Mon Mar 21 at 9:40am)
• Don't forget to turn in your in-class activities, in person or on Gradescope!
  (due Sun Mar 27 at 11:59pm)
• You can turn in a metacognition essay on Gradescope
  (due Sun Mar 27 at 11:59pm)

Homework (due May 11):

• Euler's theorem (10 points)
• Euler's phi-function* (15 points)

No projects assigned this week, but check back later, I might think of something!

Work due this week:

• Read Section 7, then answer Reading quiz 6 on Gradescope
  (due Mon Mar 14 at 9:40am)
• Don't forget to turn in your in-class activities, in person or on Gradescope!
  (due Sun Mar 20 at 11:59pm)
• You can turn in a metacognition essay on Gradescope
  (due Sun Mar 20 at 11:59pm)

Homework (due May 11):

• The divisors of an integer (15 points)
• Another multiplicative function* (5 points)

As a project this week, you can complete a short reading and an exercise on the Möbius inversion formula (20 points)

Work due this week:

• You can turn in a metacognition essay on Gradescope
  (due Sun Mar 13 at 11:59pm)

No new homework or material this week, though we will finish the Week 5 material in class.

As a project this week, you can read Section 6 of our textbook, and solve the problems given in the assignment Fermat's Little and Wilson's Theorems.

Work due this week:

• Read Section 5, then answer Reading quiz 5 on Gradescope
  (due Wed Feb 23 at 9:40am)
• Don't forget to turn in your in-class activities, in person or on Gradescope!
  (due Sun Feb 27 at 11:59pm)
• You can turn in a metacognition essay on Gradescope
  (due Sun Feb 27 at 11:59pm)

Homework (due May 11):

• The ring ZZ/nZZ (5 points) - this hw was moved over from last week
• Single linear congruences* (15 points) - grad students only required to do prob 2
• Solving multiple linear congruences* (15 points) - grad students only required to do probs 4, 5

As a project this week, consider what can happen when solving multiple linear congruences and the moduli are not relatively prime. Consider several examples showing different possibilities (up to 10 points), and then formulate a conjecture on the possible number(s) of solutions (up to 10 more points).

Work due this week:

• Read Section 4, then answer Reading quiz 4 on Gradescope
  (due Mon Feb 14 at 9:40am)
• Don't forget to turn in your in-class activities, in person or on Gradescope!
  (due Sun Feb 20 at 11:59pm)
• You can turn in a metacognition essay on Gradescope
  (due Sun Feb 20 at 11:59pm)

Homework (due May 11):

• Beginning congruences (10 points)
• Essentials of congruence proofs (10 points)
• Advanced congruence proofs* (10 points)
• The assignment "The ring ZZ/nZZ" has been moved to Week 5

No projects assigned this week, but check back later, I might think of something!

Work due this week:

• Read Section 2, then answer Reading quiz 3 on Gradescope
  (due Mon Feb 7 at 9:40am)
• Don't forget to turn in your in-class activities, in person or on Gradescope!
  (due Sun Feb 13 at 11:59pm)
• You can turn in a metacognition essay on Gradescope
  (due Sun Feb 13 at 11:59pm)

Homework (due May 11):

• Explorations on primes (5 points)
• Proofs and primes* (10 points)

Projects (due May 11):

• A short reading and exercise on the sieve of Eratosthenes (5 points)
• Some proofs on the least common multiple (15 points)

Work due this week:

• Read the rest of Section 1, then answer Reading quiz 2 on Gradescope
  (due Mon Jan 31 at 9:40am)
• Don't forget to turn in your in-class activities, in person or on Gradescope!
  (due Sun Feb 6 at 11:59pm)
• You can turn in a metacognition essay on Gradescope
  (due Sun Feb 6 at 11:59pm)

Homework (due May 11):

• The Euclidean algorithm by hand (10 points)
• Solving ax + by = d by hand (10 points)
• Implementing the Euclidean algorithm (10 points)
• Proofs, the gcd and the Euclidean algorithm* (10 points)

As a project this week, you can read Section 3 of our textbook, and solve the problems given in the assignment Solving ax + by = c completely (30 points)

Work due this week:

• Read pages 1-6 of the book, then answer Reading quiz 1 on Gradescope
  (due Mon Jan 24 at 9:40am)
• Don't forget to turn in your in-class activities, in person or on Gradescope!
  (due Sun Jan 30 at 11:59pm)
• You can turn in a metacognition essay on Gradescope
  (due Sun Jan 30 at 11:59pm)

Homework (due May 11):

• Proofs using mathematical induction (5 points)
• Proofs on divisibility and the gcd* (15 points)
• Proofs using the Division Algorithm (10 points)

No projects assigned this week, but check back later, I might think of something!

On Wednesday we had an introduction to the class and I read the syllabus at length. Please don't hesitate to send me any questions you might have about it.
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On Friday I will talk about the basic facts about numbers that we will use this semester.
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There is no homework or anything due before Monday morning.