NUMBER THEORY 1 SECTION 1 FALL 2022 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 ALMOST 1/3 OF GRADE.
TRY PRACTICE PROBLEMS EVERY DAY!
EXAMS WILL BE ANNOUNCED BELOW
PLEASE READ ALL INSTRUCTIONS ABOVE AND KEEP UP WITH CLASS.
12/13/2022 TUESDAY
Grades have been posted on banner. Please wait until it gets published.
Please read solutions for Final exam given below.
Final exam solutions
Final exam problems
Enjoy the break!
FINAL EXAM DEC 9 FRIDAY 2PM SAME ROOM
Review for final exam Wednesday 3.30 pm in usual room.
12/2/2022 FRIDAY
Test 3 problems
Solutions to Test 3
Please get started on wikipedia course if you haven't already.
TEST 3 FRIDAY DEC 2, REVIEW TOMORROW WED
SAMPLE PROBLEMS BELOW.
FINAL DEC 9, 2 TO 4PM
11/30/2022 WEDNESDAY
Please submit hw7 by friday.
Please read the solutions below. I have included the solution to the last probem we were discussing in class.
Solutions to Test 3 practice problems
homework 7 problems:
1. Chapter 21 problems 1 and 4.
2. Prove that the least non-residue mod a prime has to be a prime itself.
3. Chapter 20, 1 and 2.
11/28/2022 MONDAY
Today we talked about proving the quadratic reciprocity law (Chapter 22, 23).
Please read those two chapters.
Notes : (Refresh browser!) Quadratic congruences
homework 7 problems:
1. Chapter 21 problems 1 and 4.
2. Prove that the least non-residue mod a prime has to be a prime itself.
3. Chapter 20, 1 and 2.
11/23/2022 TUESDAY
SAMPLE PROBLEMS FOR TEST 3
On monday we will finish chapter 23.
11/22/2022 MONDAY
Today we talked about calculating quadratic residue of 2 and an application to Artin's primitive roots conjecture (Chapter 21).
Notes : (Refresh browser!) Quadratic congruences
homework 7 problems:
1. Chapter 21 problems 1 and 4.
2. Prove that the least non-residue mod a prime has to be a prime itself.
3. Chapter 20, 1 and 2.
11/18/2022 FRIDAY
Today we talked about calculating quadratic residue of 2 (Chapter 21).
Notes : (Refresh browser!) Quadratic congruences
homework 7 problems:
1. Chapter 21 problems 1 and 4.
2. Prove that the least non-residue mod a prime has to be a prime itself.
3. Chapter 20, 1 and 2.
11/17/2022 THURSDAY
Yesterday we talked about calculating quadratic residues and non-residues (Chapter 21).
I have added a section about least quadratic non-residues in the notes below.
Notes : (Refresh browser!) Quadratic congruences
homework problems:
1. Chapter 21 problems 1.
2. Prove that the least non-residue mod a prime has to be a prime itself.
11/14/2022 MONDAY
Today we talked about quadratic residues and non-residues (Chapter 20).
Notes : Quadratic congruences
homework problems: Chapter 20 problems 1, 2.
11/7/2022 MONDAY
Today we talked about primality testing using Miller-Rabin test (Chapter 19).
Notes : (Updated -- reload this page) Congruences
homework problems: Chapter 19 problem 7.
On wednesday we will start chapter 20.
11/4/2022 FRIDAY
Today we talked about primality testing and Korselt's criterion for Carmichael numbers (Chapter 19).
Notes : (Updated -- reload this page) Congruences
homework problems: Chapter 19 problems 2, 3, and 4.
On monday we will finish chapter 19.
11/3/2022 THURSDAY
Test 2 problems
solutions to Test 2
10/29/2022 SATURDAY
MONDAY CLASS ON ZOOM.
SAMPLE PROBLEMS FOR TEST 2
Yesterday we talked about the RSA cryptosystem and finding roots modulo an integer.
Notes : (Updated -- reload this page) Congruences
On monday we will review for test 2.
TEST 2 WED NOV 2. DETAILS SOON.
10/26/2022 WEDNESDAY
Today we talked about Euler's function and the RSA cryptosystem.
Notes for today: (Updated -- reload this page) Congruences
On monday we talked about the prime counting function and some open (unsolved) problems involving primes (chapter 13).
I have updated these notes from the introduction to include that discussion.
Since chapter 13 is mostly a survey, it is left as a reading assignment for you.
HW problems for today:
Chapter 16 problems 3, 5.
Chapter 17 problems 1, 3a, 5.
10/21/2022 FRIDAY
Today we talked about Carmichael numbers, Euler's function and Chinese Remainder theorem.
Notes for today: (Updated -- reload this page) Congruences
HW problems for today:
1. Let m,n be relatively prime. Show that the map from integers prime to mn to pairs of integers prime to m and n given by f(x) = (x mod m, x mod n) is a 1-1 map.
Chapter 11 problems 3,6,9.
10/19/2022 WEDNESDAY
FRIDAY'S CLASS WILL BE ON ZOOM.
Today we talked about Fermat's little theorem and primality testing.
Notes for today: (Updated -- reload this page) Congruences
HW problems for today:
1. Show that the product of the elements in any finite abelian group is (a) the identity if the group is of odd order (b) the unique element of order 1 if the group is cyclic and has even order.
Chapter 9, problem 4.
Chapter 10, problem 3.
10/17/2022 MONDAY
This week we talked more about Wilson's theorem and its generalization (Chapters 9 and 10 exercises)
Notes for today: (Updated -- reload this page) Congruences
10/14/2022 FRIDAY
This week we talked more about Fermat's little theorem and Euler's theorem. (Chapters 9 and 10)
Notes for today: (Updated -- reload this page) Congruences
10/7/2022 FRIDAY
Today we talked about exponentiation in Congruences and Fermat's little theorem. (Chapter 9)
Notes for today: (Updated -- reload this page) Congruences
HW problems from today: Chapter 9, problems 1, 2.
10/5/2022 WEDNESDAY
Today we talked about solutions of Congruences. Video link will be sent
Notes for today: (Updated -- reload this page) Congruences
HW problems from today: Chapter 8, problems 6, 9, 10.
10/3/2022 MONDAY
Test 1 problems
solutions to Test 1
TEST I ON MONDAY OCT 3. WILL COVER CHAPTERS 1 TO 8 and 12.
PLEASE STUDY CLASS NOTES, HOMEWORK PROBLEMS AND TEXTBOOK.
THERE WILL BE QUESTIONS ON THEORY, QUESTIONS ON EXAMPLES, AND COMPUTATIONAL QUESTIONS.
Sample questions
9/30/2022 FRIDAY
Brief solutions to HW3
9/28/2022 WEDNESDAY
Please get started on wikipedia course if you haven't already.
Today we talked about Congruences and Modular Arithmetic.
Notes for today: Congruences
Practice problem for test from today: Chapter 8, problems 1, 2, 3, 4, 5.
9/26/2022 MONDAY
Please get started on wikipedia course if you haven't already.
Today we talked about Unique Factorization.
Notes for today: Unique Factorization
Practice problem for test from today: Chapter 7, problems 1, 2, 3, 4, 6.
9/25/2022 SUNDAY
Brief solutions to HW2
9/21/2022 WEDNESDAY
Please get started on wikipedia course if you haven't already.
Today we started talking about GCD.
Notes for today: (Updated-- please reload) GCD
Homework problem for today: See last page of notes above.
9/19/2022 MONDAY
Please get started on wikipedia course if you haven't already.
Today we started talking about GCD.
Notes for today: GCD
Homework problem for today: chapter 5: 5.1, 5.3, 5.4, 5.5(a,b,c).
9/16/22 FRIDAY
Today we discussed some of the homework problems and also problem 3.4 from book on rational points on the elliptic curve.
Notes and link to video have been emailed.
9/14/2022 WEDNESDAY
Please get started on wikipedia course if you haven't already.
Today we finished talking about Pythagorean triples.
(Updated -- reload browser) Notes for today: Pythagorean Triples
No Homework problem for today, but please try as many problems from chapters 3 and 4 as possible.
We will go over some of them on friday.
Your homework from past week is due friday. See below for list.
9/12/2022 MONDAY
Please get started on wikipedia course if you haven't already.
Today we talked more about Pythagorean triples.
Video "Pi hiding in prime irregularities" including the count of lattice points on circles
We will cover the math from this video as we go along.
(Updated -- reload browser) Notes for today: Pythagorean Triples
Homework problem for today:
Write 85 as sum of squares in two ways using the method we discussed.
Try it before you figure out what the squares are!
The five problems assigned so far (2.1a, 2.2, 2.5, 2.6 and the one above) are due on friday.
9/9/2022 FRIDAY, 11pm
Please get started on wikipedia course if you haven't already.
Today we talked more about Pythagorean triples.
(Updated -- reload browser) Notes on that: Pythagorean Triples
Homework problem for today:
Problems 2.5 and 2.6 from book:
2.5.About triangular numbers and Pythagorean triples.
2.6. About triples of form $a, b, a+2.$.
9/9/2022 FRIDAY, 2pm
Brief solutions to HW1
9/7/2022 WEDNESDAY
Please get started on wikipedia course if you haven't already.
Today we talked about Pythagorean triples.
Notes on that: Pythagorean Triples
Homework problem for today:
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.
9/1/2022 THURSDAY
Some proofs of the infinitude of primes
Nice article by K. Conrad on patterns in primes, including polynomials which take prime values infinitely often
8/31/2022 WEDNESDAY
Today we talked about Euler's proof of infinitude of primes.
(Updated -- Reload page!) Notes from today: Introduction to Number Theory
The infinitude of primes is discussed in chapter 12.
We also started on Chapter 2, Pythagorean Triples.
Notes on that: Pythagorean Triples
HW1 (total 5 problems, see below) due on friday this week.
8/29/2022 MONDAY
Today we talked infinitude of primes of form 4n-1.
(Updated -- Reload page!) Notes from today: Introduction to Number Theory
The infinitude of primes is discussed in chapter 12.
HW problems for today:
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.
HW1 (total 5 problems) due on friday this week.
8/26/2022 FRIDAY
Today we talked more about primitive roots and Artin's primitive roots conjecture.
As we learn more, we will discuss more about it.
(Updated) Notes from today: Introduction to Number Theory
We also went over Euclid's proof of the infinitude of primes.
HW problems: 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 friday next week.
8/24/2022 WEDNESDAY
Today we had a very elementary look at Artin's primitive roots conjecture.
As we learn more, we will discuss more about it.
(Updated) Notes from today: Introduction to Number Theory
We will start talking about the prime number sequence on friday.
HW problem: Find the next prime after 7 for which 10 is a primitive root (In the notes I go upto 13 and show that 11 and 13 are not).
8/22/2022 MONDAY
Today we had introductions of each other and to number theory.
Notes from today: Introduction to Number Theory
We will continue with the introduction wednesday. Please continue to read the introduction to the book.
8/19/22 Friday
PLEASE REGISTER FOR WIKIPEDIA ASSIGNMENT (SEE EMAIL SENT A FEW WEEKS EARLIER) IF YOU HAVEN'T.
8/10/2022 WEDNESDAY
Please study following to prepare for class:
INTRODUCTION TO TEXT BOOK