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Theory of Numbers

Arizona State University, Spring 2023

When: Tuesdays and Thursdays 1:30–2:45pm

Where: WXLR A306

Instructor: Zilin Jiang ([email protected])

Office hours: Tuesday and Thursday 2:50pm–4pm, in WXLR A839

Teaching assistant: Steven Ruiz ([email protected])

TA MCC office hours: Tuesday and Thursday 11:30am–1pm, in WXLR A303 (MCC)

Course description​

This course is an elementary introduction to number theory. Topics covered include primes numbers, unique factorization theorem, congruences, Diophantine equations, primitive roots, and quadratic reciprocity theorem.

Prerequisites: Students are assumed to have been introduced to elementary proof techniques and mathematical reasoning. These modest prerequisites are typically developed at the late sophomore or early junior level.

Primary reference: Andrew Granville. Number Theory Revealed: An Introduction. American Mathematical Society, 2019.

Remark: The book by Granville is not required. The class will be self-contained, and the homework is not related to the textbook.

Onboarding:

  • Please fill out the pre-semester questionnaire.
  • (Optional) Learn how to use LaTeX (I recommend the tutorials from Overleaf).

Asking questions: For class-related questions, it is best to come to office hours. Another possibility: ask them on Slack to get help from your peers. Administrative questions too can be asked there, especially if other students might be able to answer them or might learn from the answer.

Grading​

Homework 40% + Two Midterms 15% each + Final 30% + Bonus 10%

Assignments​

You are encouraged to type your submissions in LaTeX. All steps should be justified. You are strongly encouraged to do as much homework as possible individually; you will gain the most out of the course this way.

Sources and collaboration policy. Collaboration and use of external sources are permitted, but must be fully acknowledged and cited. Collaboration may involve only discussion; all the writing must be done individually. Failure to do so will be treated as cheating (see What is Academic Integrity?).

Late policy and grade appeal policy. In general no late homework will be accepted unless there is a genuine emergency backed up by official documents. No grade appeal will be considered after one week of the posting of the grade for any assignment.

There will be 7 assignments.

  1. Assignment 1 due on Jan 19.
  2. Assignment 2 due on Feb 2.
  3. Assignment 3 due on Feb 21.
  4. Assignment 4 due on Mar 14.
  5. Assignment 5 due on Mar 30.
  6. Assignment 6 due on Apr 13 (extended to Apr 16).
  7. Assignment 7 due on Apr 27.

Exams​

No make-up test will be given unless a student has notified the instructor before the test is given.

  • Midterm 1: Feb 9, 1:30–2:45pm.
  • Midterm 2: Mar 23, 1:30–2:45pm.
  • Final: May 4, 12:10–2:00pm.

Bonus​

You will get up to 10% as a bonus for shooting a short film about number theory. The short film should be up to 10 minutes long (exceptions can be made upon approval from the instructor). Here is the timeline for the bonus project:

  1. Form a team of up to 5 people, and sign up for the bonus by Mar 3.
  2. Prepare a script and/or a storyboard, and get approved by Mar 14.
  3. If necessary, modify the script and/or the storyboard, and get approved by Mar 28.
  4. Filming and post-production, and get feedback by Apr 11.
  5. Submit the final short film by Apr 25.

Schedule​

Week 1

  • Jan 10. Induction, Fibonacci numbers, sums of powers of integers, divisibilities.
  • Jan 12. Division with remainder, greatest common divisor (gcd), Euclidean algorithm.

Week 2

  • Jan 17. Ideal generated by a and b is the same as the ideal generated by gcd(a, b), Bézout's identity and its consequences.
  • Jan 19. General solution to ax + by = c, reversing the Euclidean algorithm to find a special solution to ax + by = gcd(a, b), least common multiple and its properties.

Week 3

  • Jan 24. Generalize gcd and lcm to more integers, Bézout's identity extended to more than two integers, a common divisor divides the gcd, and the lcm is a multiple of a common multiple.
  • Jan 26. Congruence modulo m, basic arithmetic properties, tests for divisibility.

Week 4

  • Jan 31. Periodicity of Fibonacci numbers modulo m, the pigeonhole principle, prime and composite, factorization into primes.
  • Feb 2. Fundamental theorem of arithmetic, Euclid's lemma, divisor function τ(n).

Week 5

  • Feb 7. Looking through the lens of the fundamental theorem of arithmetic, irrationality of √d, the rational root theorem.
  • Feb 9. Midterm 1.

Week 6

  • Feb 14. The rational root theorem for monic polynomials, dividing in congruences, a card trick based on congruences to several moduli.
  • Feb 16. The Chinese remainder theorem.

Week 7

  • Feb 21. When the moduli are not necessarily pairwise coprime; square roots of 1 modulo a power of an odd prime.
  • Feb 23. Square roots of 1 modulo an odd number; multiplicative and completely multiplicative functions; Euler's totient function is multiplicative.

Week 8

  • Feb 28. Formula for Euler's totient function φ; sum of φ(d) over all positive divisors of n equals n; perfect numbers; formula for the sum σ(n) of all positive divisors of n; sum of f(d) over all positive divisors of n is multiplicative as long as f is.
  • Mar 2. Breaking down math videos — a presentation on the bonus project by Jae Joiner.

Week 9

  • Mar 7. No class (spring break).
  • Mar 9. No class (spring break).

Week 10

  • Mar 14. Two formulas involving π, convolution of arithmetic functions, convolution of two multiplicative functions is still multiplicative, the Möbius function, and the Möbius inversion formula.
  • Mar 16. Infinitude of primes and primes congruent to 2 modulo 3, Dirichlet's theorem on arithmetic progressions, the prime number theorem, binomial coefficients and their properties, Kummer's theorem.

Week 11

  • Mar 21. Chebyshev's theorem on the prime counting function π(x).
  • Mar 23. Midterm 2.

Week 12

  • Mar 28. Bertrand's postulate, the Sylvester–Schur theorem, Pythagorean triples, the characterization of the primitive Pythagorean triples.
  • Mar 30. Fermat's last theorem, Fermat's infinite descent proof of Fermat's last theorem for n = 4, the equation x^4 - y^4 = z^2 has no solutions in positive integers, power residue, order of a modulo m.

Week 13

  • Apr 4. Euler's theorem, Fermat's little theorem, Lagrange's theorem on the roots modulo p of a polynomial with integer coefficients.
  • Apr 6. Wilson's theorem, number of elements of a given order, primitive roots and their applications.

Week 14

  • Apr 11. A criterion for primitive roots, Carmichael numbers, divisibility test again.
  • Apr 13. Quadratic residues and quadratic non-residues, the Legendre symbol, Euler's criterion, (-1/p), Gauss's lemma, (2/p).

Week 15

  • Apr 18. The law of quadratic reciprocity.
  • Apr 20. The Jacobi symbols, sum of two squares.

Week 16

  • Apr 25. Private key scheme, the RSA public key scheme, the AKS primality test.
  • Apr 27. A presentation on semester highlights and short film exhibition.