NUMBER THEORY 2 SECTION 1 SPRING 2025 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.

HOMEWORK IS ALMOST 1/3 OF GRADE.

EXAMS WILL BE ANNOUNCED BELOW

NOTES AND EXAM SOLUTIONS FROM SPRING 2023.


PLEASE READ ALL INSTRUCTIONS ABOVE AND KEEP UP WITH CLASS.




4/24/25 Thursday
Yesterday Dr. Babei gave a guest lecture on L-functions of number fields and elliptic curves.

HW5 problems (with problems from elliptic curves):
1. 31.1 (four solutions to x^2-5y^2=1).
2. 32.2 (solutions to x^2-22y^2=1).
3. (adapted from 33.2) Find a rational number x/20 that approximates the golden ratio within 1/20.
4. 33.3 (golden ratio and continued fractions).
5. 34.1a (when does negative Pell has solution).
6. 35.9 (units in Q(sqrt(3)).
7. 36.6 (a to g) (factorization in Q(sqrt(-5)).
8, 9, 10: 41.1, 41.2 and 42.2 (rational points on elliptic curves with RHS x^3+17, x^3+x-1, x^3+1 respectively).

4/17/25 Thursday
HW5 problems:
1. 31.1 (four solutions to x^2-5y^2=1).
2. 32.2 (solutions to x^2-22y^2=1).
3. (adapted from 33.2) Find a rational number x/20 that approximates the golden ratio within 1/20.
4. 33.3 (golden ratio and continued fractions).
5. 34.1a (when does negative Pell has solution).
6. 35.9 (units in Q(sqrt(3)).
7. 36.6 (a to g) (factorization in Q(sqrt(-5)).

We have been talking about diophantine approximation and Pell's equations.
Notes on diophantine equations.
We also started talking about elliptic curves.
Notes on elliptic curves.


4/10/25 Thursday
test 2 problems.
Solutions for test 2.

4/4/25 Friday
For TEST 2 on wed 4/9 please study notes, hw problems and these test 2 practice problems.
Solutions for test 2 practice problems.

We have been talking about diophantine problems, such as Fermat's and Pell's equations.
Notes on diophantine equations.
We also talked about structure of quadratic number fields.
Read these notes by Keith Conrad for more.
Also check out these notes by Michel Waldschmidt on Pell's equation.


3/19/25 Wednesday
Today we concluded discussion of Euler function and using a result on Euler's Totient Function proved cyclicity of multiplicative group of finite field of prime number order.
Notes on Euler Function and Primitive Roots (Refresh!).
HW4 problems: Due Monday 3/31.
Problem 1: 24.6 (Showing that any integer modulo which there is a square root for -1 is a sum of two squares.
Problems 2 to 5: 25.1, 25.2, 25.6 (about numbers that are sums of squares).
Problem 6: Prove that the set of arithmetic functions which are nonzero at 1 form a group under Dirichlet convolution.
Problem 7: Prove that the set of integers k < or equal to n such that gcd(k,n) = d has phi(n/d) elements.
Problems 8 and 9: 27.1, 27.2 (in Euler phi function chapter).
Problems 10, 11, 12 : (Primitive roots chapter) 28.1, 28.3, 28.5.
Problem 13: 29.6 (El-Gamal cryptosystem).
Problems 14, 15: 35.7, 35.8.


3/12/25 Wednesday
Today we concluded discussion of sums of squares, and started on Euler's Totient Function.
Notes on Fermat numbers and sums of squares (Refresh!).
HW4 problems:
Problem 1: 24.6 (Showing that any integer modulo which there is a square root for -1 is a sum of two squares.
Problems 2 to 5: 25.1, 25.2, 25.6 (about numbers that are sums of squares).


2/26/25 Wednesday
test 1 problems.

test 1 solutions.


2/20/25 Thursday
For TEST 1 on wed 2/26 please study notes, hw problems and these test 1 practice problems.

On wednesday we started talking about Fermat numbers and sums of squares.
Notes on Fermat numbers and sums of squares.
HW3 problems:
Problem 1: Show that if a is a square mod b, with a, b odd, then the Jacobi symbol of a mod b is 1 but converse is not true.
Problems 2 to 3: 23.4 (another proof for quadratic residue of 2), 23.5 (Pick's theorem for triangle).
Problems 4 to 6: 14.1, 14.2, 14.3 (problems on Fermat numbers and (3^n-1)/2).
Problem 7: Prove Pepin's test see notes above.

2/12/25 Wednesday
On monday we finished proof of quadratic reciprocity law.
Today we talked about Pick's theorem (see zoom whiteboard) which counts number of lattice points on a simple polygon all of whose vertices are integer lattice points.
HW3 problems:
Problem 1: Show that if a is a square mod b, with a, b odd, then the Jacobi symbol of a mod b is 1 but converse is not true.
Problems 2 to 3: 23.4 (another proof for quadratic residue of 2), 23.5 (Pick's theorem for triangle).

2/5/25 Wednesday
We talked about Frobenius automorphism and quadratic reciprocity.
Notes on Quadratic Congruences (Reload!).
HW2 problems:
Problems 1 to 5: 21.1, 21.2, 21.4, 21.5, 21.6.
Problems 6 to 11: 22.1, 22.2, 22.3, 22.4, 22.6, 22.10.

2/3/25 Monday
We continued same topics as wednesday.
Notes on Quadratic Congruences (Reload!).
HW2 problems:
Problems 1 to 5: 21.1, 21.2, 21.4, 21.5, 21.6.
Problems 6 to 11: 22.1, 22.2, 22.3, 22.4, 22.6, 22.10.

1/29/25 Wednesday
We continued quadratic congruences, talked about square root of 2 mod a prime.
Notes on Quadratic Congruences (Reload!).
HW2 problems:
Problems 1 to 5: 21.1, 21.2, 21.4, 21.5, 21.6.

1/27/25 Wednesday
We started quadratic congruences, talked about Euler's criterion and square root of -1 mod a prime.
Notes on Quadratic Congruences (Reload!).
HW2 problems:
Problems 1, 2: 21.1, 21.2.

1/22/25 Wednesday
We finished talking about primality tests and started quadratic congruences.
Notes on Congruences (Reload!).
Notes on Quadratic Congruences (Reload!).
HW1 problems:
Problems 1 to 4: 19.1, 19.2, 19.4, 19.7.
Problem 5: Show that, for odd n, (x+a)^n = x^n + a (mod n) for all a relatively prime to n iff n is prime.
Problem 6: Let S be the subgroup of residues in F the multiplicative group mod an odd prime p.
F/S is a group of order 2. Show that any group of order 2 is isomorphic to {1,-1} under multiplication.
Show that the isomorphism F/S -> {1,-1} maps S to 1 and Sa to -1 where a is any nonresidue. Hence it is nothing but the map taking x in F to its Legendre symbol (a/p).
This also proves the multiplicativity of Legendre symbol (including the fact that NR x NR = QR).

Problem 7: 20.2 a-d.

1/15/25 Wednesday
We finished talking about Carmichael numbers and Korselt's criterion, and started primality tests.
Notes on Congruences (Reload!).
HW1 problems: 19.1, 19.2, 19.4, 19.7.

1/14/25 Tuesday
We reviewed Carmichael numbers and Korselt's criterion.
Notes on Congruences (Reload!).
HW1 problems today: 19.1, 19.2.