Math 105 (Fall 2025)

October 06: 

October 03: We derived some more consequences of Bezout's lemma: if aX + bY = c has a solution, then c must be divisible by gcd(a, b); if a prime p divides the product ab, then p must divide either a or p must divide b. By induction, the latter also gives that if p divides a product a_1 a_2 a_3 ... a_r, then p must divide one of the a_i. We then stated the fundamental theorem of arithmetic, and began proving it by induction on the number of prime factors in a factorization.

October 01: We saw the key facts which make the Euclidean algorithm work: gcd(a, 0) = a and gcd(a + bq, b) = gcd(a, b) for all integers a, b, and q. We then saw how to use the Euclidean algorithm to obtain Bezout's lemma: given integers a, b, there exists integers r, s such that gcd(a, b) = ra + sb. The integers r and s are often called Bezout coefficients for a and b; they are not unique. We showed that every common divisor of a and b divides gcd(a, b).

September 29: Defined prime integers and divisors. Proved
(i) every integer > 1 has a prime divisor,
(ii) there are infinitely many primes, and
(iii) every integer > 1 has a prime factorization [we do not know yet that this prime factorization is unique!]. 

Then defined the greatest common divisor of two integers and ran the Euclidean algorithm on 54 and 21. Make sure to check carefully that for integers a, b, c, r, s, if a divides both b and c, then a divides rb + sc, and if a divides b and b divides c, then a divides c. 

September 25: Introduction to the course; discussed integral solutions of y^2 = 2^x + 3 in detail.

Lecture Time: MWF 11am to 11:50am
Lecture Location: Peterson Hall Room 102
Textbook: An Introduction to Number Theory, by Harold M. Stark. The MIT Press; Seventh Printing edition. 

Discussion Section:
Section A01: Tu 11am to 11:50am at HSS 4025
Section A02: Tu 12pm to 12:50pm at HSS 4025

Course Personnel:
Instructor: Karthik Ganapathy Venkitachalam
Teaching Assistant: Nathan Wenger
Note: To contact us, please use Piazza. The system is highly catered to getting you help fast and efficiently from classmates, the TA, and myself. Rather than emailing questions to the teaching staff, I encourage you to post your questions on Piazza.

Office Hours:
Karthik Ganapathy: W 3:30-5:00pm and F 9:30-11:00am at AP&M1230
Nathan Wenger: Tu 1:30-3:30pm in HSS 5029

Homework:
Homework assignments will be posted on Canvas almost every week. Normally you will have one week to submit your completed homework via Gradescope. Your lowest homework assignment will be dropped at the end of the course when computing your grade.

Exam:
There will be two midterm exams and one final exam. No makeup exams will be given in this course.
Note: It is standard Math Department practice to utilize different versions of exams, both within each lecture's exam, and between lectures whose exams are at different times.

Midterm I: 7pm-8pm on October 20 in Warren Lecture Hall 2111.
Midterm II: 7pm-8pm on November 17 in Warren Lecture Hall 2111.
Final exam: 11:30am-2:30pm on December 09 at PETERSON 102.

Regrades: Regrade requests will be made using the built-in regrade request feature in Gradescope. There will be a limited window of time after the exams are made available during which the regrade request feature will be active. This time window will be announced when the exam scores are released. Only the regrade requests sent within the time window will be considered.

Grading Policy:

The grades will be determined as the best of: 

a) 20% Homework, 20% Midterm Exam I, 20% Midterm Exam II, 40% Final Exam; OR

b) 25% Homework, 25% Best Midterm Exam, 50% Final Exam.

Attendance Policy: You are expected to attend every session of the lectures and discussion section you are enrolled in. Students who do not attend class rarely succeed in this course.  If you must miss class for any reason, it is your responsibility to catch up on the material. You are also responsible for the information given in any announcements made during class. To help you with this, I will record every lectures in the event you are unable to attend. Lecture recordings can be found at UCSD Podcast.

Academic Integrity: Academic dishonesty is considered a serious offense at UCSD. Students caught cheating will face an administrative sanction which may include suspension or expulsion from the university. It is in your best interest to take pride in your work maintain your academic integrity.

Accommodations: Students requesting accommodations for this course due to a disability must provide a current Authorization for Accommodation (AFA) letter issued by the Office for Students with Disabilities (https://osd.ucsd.edu/.) Students are required to discuss accommodation arrangements with instructors and OSD liaisons in the department in advance of any exams or assignments.

Here are some resources from other UCSD instructors that you may find useful:
1) Cristian Popescu's course website from Winter 2021
2) Aaron Pollack's course website from Winter 2022