NUMBER THEORY 1 SECTION 1 FALL 2024 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 FALL 2022.
PLEASE READ ALL INSTRUCTIONS ABOVE AND KEEP UP WITH CLASS.
THURSDAY OFFICE HOURS ON ZOOM.
12/10/24 Tuesday
Final exam Solutions
Final exam Problems
12/6/24 Friday
FINAL EXAM RESCHEDULED DEC 9 FRI 10AM IN MY OFFICE ASB-B 219.
Practice problems Solutions
Practice Problems for Final
11/25/2024 MONDAY
FINAL EXAM DEC 6 FRI 2PM IN MY OFFICE ASB-B 219.
We talked about HW5 and finding roots and orders of elements mod m.
Notes on Congruences (Reload!).
Please read chapters 12 and 18.
11/20/2024 WEDNESDAY
Next monday class will be on zoom.
We talked about HW5 and method of successive squaring.
Notes on Congruences (Reload!).
11/18/2024 MONDAY
We talked about Mersenne primes and perfect numbers.
Notes on Congruences (Reload!).
We also proved that if m is the highest exponent of a prime p in the factorization of n! then m is smaller than n/(p-1).
11/14/2024 THURSDAY
We talked about Euler's formula and chinese remainder theorem wednesday.
Notes on Congruences (Reload!).
HW5 problems: chapter 11 problems 1, 2, 3, 5, 6, 9, 10, 11, 12, 13.
11/7/24 Thursday
Test 2 Solutions
test 2 Problems
11/1/2024 FRIDAY
TEST 2 WEDNESDAY NOV 6.
Solutions for sample problems for Test 2.
For the test please study sample problems, class notes, homework problems, old tests (from fall 2022). Note that test 2 in fall 2022 covered a few more topics.
Please study these sample problems for Test 2.
We will go over as many of them monday as possible.
10/31/2024 THURSDAY
We continued with congruences wednesday.
Notes on Congruences (Reload!).
10/25/2024 FRIDAY
HW4 problems:
7.1, 7.2, 7.4, 8.1, 8.2, 8.3, 8.4, 8.9, 8.10.
In an abelian group show that x^2 = e where x is the product of all the elements in the group.
Using this show that P^2 = 1 where P is the product of all non-zero remainders mod a prime p.
We continued with congruences wednesday.
Notes on Congruences.
10/21/2024 MONDAY
Please submit HW3 and Wikipedia topic ASAP.
We started congruences today.
Notes on Congruences.
10/16/2024 WEDNESDAY
TEST 2 MONDAY NOV 4.
HW3 and Wikipedia topic due Monday Oct 21.
10/11/2024 FRIDAY
Notes on Unique Factorization (Updated -- please reload!).
On wednesday we proved that in a PID you have factorization into irreducibles.
More HW3 problems: (HW3 DUE WEDNESDAY)
5. Show that an element u = a+ib in Z[i] is a unit iff a^2+b^2 = 1. Find all the units in this ring.
6. Show that a+ib in Z[i] is irreducible if a^2+b^2 is a prime number.
7. Let R be a Euclidean domain that is not a field. Show that there is a in R such that
every coset in R/(a) is of the form (a)+r where r is either 0 or a unit of R.
8. A GCD-domain is an integral domain in which any two elements a, b have a GCD: an element g such that g divides a and b, AND if d divides a and b then d divides g.
Show that irreducible elements in any GCD domain are also prime elements.
9. Show that an integral domain that is Noetherian (see notes) has factorization into irreducibles (not necessarily unique).
10. Show that Noetherian domains which have GCD property are UFD's.
11. Z[x] is known to be Noetherian (see notes). Show that Z[x] is also a GCD domain, and hence a UFD (note that it is neither Euclidean nor a PID).
Notes on Unique Factorization.
Today we talked about Euclidean domains, PIDs, and primes and irreducibles.
Some HW3 problems:
1. Show that Z[x] is not a PID, where Z is the ring of integers.
2. We showed in class that Z[x] cannot be a Euclidean domain because Z is not a field.
Explain why this makes Euclidean algorithm not work.
3. Show that Q[x]/(f(x)) is a field whenever f(x) is an irreducible polynomial in the field
of rational numbers Q.
4. If f(x) = x^2+x+1 in (3) find the image of x^3+2 mod f(x) and its inverse.
Test 1 Solutions
test 1 Problems
Test I will be on wednesday Sep 23. It will cover upto chapter 6.
REVIEW IN CLASS ON MONDAY.
9/23/2024 MONDAY
Problem 16: Primes in Pythagorean Triples.
For the test please study sample problems, class notes, homework problems, old tests (from fall 2022). Note that test 1 in fall 2022 covered a few more topics.
9/17/2024 TUESDAY
Please study these sample problems for Test 1.
We will go over as many of them monday as possible.
For the test please study class notes, homework problems, old tests (from fall 2022). Note that test 1 in fall 2022 covered a few more topics.
9/11/2024 WEDNESDAY
We started talking about GCD.
Notes for today and next Monday
HW2 problems: Problems 5.1 and 5.3.
Please do Problem 2.4 from book using Gaussian integers
Problems 3.3, 3.4 and 3.5 from Chapter 3. In 3.5 part (f) is hard-- but you can give it a try just for fun.
9/9/2024 MONDAY
We finished the parametrization of Pythagorean triples using unit circle. Talked about the trigonometric proof and the proof using Gaussian integers.
Pythagorean Triples(Revised from fall 22)
HW2 problems: Please do Problem 2.4 from book using Gaussian integers
Problems 3.3, 3.4 and 3.5 from Chapter 3. In 3.5 part (f) is hard-- but you can give it a try just for fun.
9/4/2024 WEDNESDAY
We described the parametrization of Pythagorean triples using unit circle. Please read the notes because I have added some things to what we talked about in class.
Pythagorean Triples(Revised from fall 22)
Please try Problem 2.8 from book (just for fun!)
8/28/2024 WEDNESDAY
We described the parametrization of Pythagorean triples. Please read the notes because I have added some things to what we talked about in class.
Pythagorean Triples(Revised from fall 22)
Homework problems:
Problems 2.5 and 2.6 from book. All of the homework problems given so far are due next wednesday.
8/27/2024 TUESDAY
Yesterday we finished the intro and started on Pythagorean triples. Please read the notes because I have added some things to what we talked about in class.
Updated notes (please refresh!)
INTRODUCTION TO NUMBER THEORY (Revised from fall 22)
Homework problems:
Problems 2.1a and 2.2 from book:
2.1. (a) We showed that in any primitive Pythagorean triple (a, b, c), either a or b is even. Use the same sort of argument to show that either a or b must be a multiple of 3.
2.2. A nonzero integer d is said to divide an integer m if m = dk for some number k. Show that if d divides both m and n, then d also divides m-n and m + n.
8/21/2024 WEDNESDAY
HW problems for today: (see notes below for what we did in class).
1. We saw that product of numbers of form 4n+1 is also of form 4n+1. Same is not always true for numbers of form 4n-1.
When does product of such numbers also result in a number of form 4n-1?
2. (problem 12.2, part 1) Use the ideas from the case of 4n-1 (or 4n+3) to show that there are infinitely many primes of form 6n+5.
3. Try problems 1.3 and 1.4 after chapter 1 of the text.
If you don't have the text you can find the chapter on Prof. Silverman's website.
More hw will be assigned monday and they are all due on monday following week.
8/19/2024 MONDAY
Notes from today:
INTRODUCTION TO NUMBER THEORY (Revised from fall 22)
8/16/2024 FRIDAY
Please study following to prepare for class:
INTRODUCTION TO TEXT BOOK