NUMBER THEORY 2 SECTION 1 SPRING 2023 UPDATE PAGE

SITARAMAN

Here you will see information on what was covered in class, upcoming quizzes, and solutions to exams
CHECK HERE EVERYDAY !

COURSE PAGE WITH SYLLABUS, GRADING SCHEME, ETC.,

SCROLL DOWN BELOW TO SEE SOLUTIONS OF TESTS AND QUIZZES FROM THIS COURSE

SYLLABUS

Will try to stick to this syllabus as much as possible.

OLD FINAL EXAMS

HOMEWORK IS 40 PERCENT OF GRADE.

TESTS AND QUIZZES FROM FALL 2022

TRY PRACTICE PROBLEMS EVERY DAY!
EXAMS WILL BE ANNOUNCED BELOW

1/30/2023 MONDAY
We discussed proof by induction and went over a homework problem concerning sums of squares.
Notes (updatd--refresh browser) : Quadratic congruences, Fermat numbers and Sums of Squares

Homework problem: HW2, Problems 5: Chapter 26 problems 1.

1/27/2023 FRIDAY
We finished talking about when a prime is a sum of two squares.
Notes (updatd--refresh browser) : Quadratic congruences, Fermat numbers and Sums of Squares

Homework problem: HW2, Problems 2,3,4: Chapter 25 problems 4,5,6.

1/23/2023 MONDAY
We have been talking about when a prime is a sum of two squares..
\We also discussed sums of form n^2+1 and a^2+b^2 and how quadratic reciprocity is applied.
today we talked about Gaussian integers and prime and irreducible elements.
Notes : Quadratic congruences, Fermat numbers and Sums of Squares

Here is a nice article about the different proofs of the sum of squares theorem.
Homework problem: HW2, Problem 1: Show that 2 is irreducible but not prime in the ring of numbers of the form a+bu where u is sqrt(5) times i (or sqrt(-5)) and a, b are just integers.

1/20/2023 FRIDAY
We have been talking about when a prime is a sum of two squares..
We also discussed sums of form n^2+1 and a^2+b^2 and how quadratic reciprocity is applied.
We also talked about Gaussian integers.
Notes : Quadratic congruences, Fermat numbers and Sums of Squares

Homework problem: HW1, Problems 4 and 5: Chapter 24, problems 2 and 3.

1/13/2023 FRIDAY
We concluded talking about Lucas' theorem: If a prime p divides F_n for n > 1 then p is 1 mod 2^(n+2).
We also discussed sums of form n^2+1 and a^2+b^2 and how quadratic reciprocity is applied.
Notes : Quadratic congruences, Fermat numbers and Sums of Squares

Homework problem: HW1, Problem 3: For the primes 41 and 73 find k such that they divide k^2+1, using the procedure described in class.

1/12/2023 THURSDAY
Yesterday we looked at some applications of quadratic reciprocity law.
In particular we talked about the Fermat numbers.
We continued talking about Lucas' theorem: If a prime p divides F_n for n > 1 then p is 1 mod 2^(n+2).

Notes : Quadratic congruences and Fermat numbers

Homework problems (there are 2 of them) are in bold, inside the notes.

1/10/2023 TUESDAY
Today we reviewed the quadratic reciprocity law (Chapter 22, 23).