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
PLEASE READ ALL INSTRUCTIONS ABOVE AND KEEP UP WITH CLASS.
5/10/2023 FRIDAY
Final exam problems
Final exam solutions
5/5/2023 FRIDAY
Test 3 problems
Test 3 solutions
4/24/2023 MONDAY
We have been getting a bird's eye view of elliptic curves.
Notes (updatd--refresh browser) : A brief intro to elliptic curves
4/14/2023 FRIDAY
We talked about Dirichlet's diophantine approximation theorem.
Notes (updatd--refresh browser) : Diophantine approximation and Pell's equation (see last section)
HW5 problems (due next friday): 36.7, 33.2a, 33.3, 34.1, 34.2.
4/6/2023 THURSDAY
Yesterday talked about using factorization of Gaussian integers to count number of ways to write a number as sums of squares.
Notes (updatd--refresh browser) : Quadratic congruences, Fermat numbers and Sums of Squares
Test 2 problems
Test 2 solutions
TEST 2 on APRIL 3 MONDAY
4/1/2023 SATURDAY
More about numbers that are pentagonal, square, and triangular
3/31/2023 FRIDAY
Test 2 practice problems
Test 2 practice problem solutions
3/23/2023 THURSDAY
We talked about properties of Gaussian integers such as units and prime elements.
Notes (updatd--refresh browser) : Quadratic congruences, Fermat numbers and Sums of Squares
3/15/2023 WEDNESDAY
We discussed Pell's equation and Units in the integer rings of quadratic fields.
Notes : (updated -- please refresh): Diophantine equations especially Fermat's last Theorem.
The nice paper by Michel Waldschmidt about Pell's equation
HW 4 (worth 40 points), due next friday
Chapter 31: 1, 3, 4
Chapter 32:1, 2, 4
Chapter 35: 4, 6, 7, 8,9, 11
3/2/2023 THURSDAY
We discussed Pell's equation and Square triangular numbers.
Notes : (updated -- please refresh): Diophantine equations especially Fermat's last Theorem.
TEST I problems.
TEST I solutions.
2/23/2023 THURSDAY
We discussed diophantine equations especially Fermat's last Theorem.
TEST I will be monday. Please study notes, homework, and these practice problems.
Notes : Diophantine equations especially Fermat's last Theorem.
2/13/2023 MONDAY
We discussed some applications of cyclicity of the multiplicative group mod p (existence of primitive root), including discrete log problem and the El Gamal cryptosystem based on it.
UPDATED (please refresh) Notes : Euler Function summed over the divisors of a number, primitive root.
2/8/2023 WEDNESDAY
We discussed the proof of cyclicity of the multiplicative group mod p (existence of primitive root).
Notes : Euler Function summed over the divisors of a number, primitive root..
Homework problem: HW3, Problems 2,3,4,5: Chapter 28 problems 1,2, 4, 5.
2/1/2023 MONDAY
We discussed which numbers can be hypotenuses of primitive Pythagorean triple triangles and discussed the sum of Euler phi function over divisors.
Notes : Euler Function summed over the divisors of a number.
Homework problem: HW3, Problem 1: Chapter 27 problem 2.
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, Problem 5: Chapter 26 problem 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).
Please read those two chapters.
We talked about Fermat numbers F_n and this problem: If a prime p divides F_n for n > 1 then p is 1 mod 2^(n+2).