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Current Update:
The solutions to Problem Set 11 are now available. 

I will have office hours as usual on Thursday, Dec. 11.  During finals week, Sawyer will have an office hour on Tues, Dec. 16 from 1-2, and I will hold a session on Wed., Dec. 17 from 2 - 3:30.

You may pick up your graded PSets during office hours (or we can arrange another time if needed).

You may now download review problems for the Final Exam.

MATH 18.781 - Fall 2008

Introduction to the Theory of Numbers

This is an introductory course in number theory at the undergraduate level.  Topics will include divisibility, greatest common divisors, the Euclidean algorithm, the Fundamental Theorem of Arithmetic, the Chinese Remainder Theorem, Hensel's Lemma, Legendre symbols, quadratic reciprocity, simple continued fractions, infinite continued fractions, and Farey fractions.  There are no official prerequisites, but abstract algebra (18.700) is very helpful.

Course Information

Scheduled Time
Lectures TR 1:00
Office Hours (Karl Mahlburg) R 3:00 - 4:00 2-173
Office Hours (Sawyer Tabony)
M 2:00 - 3:00

Textbooks (recommended) I. Niven, H. Zuckerman, H. Montgomery, An Introduction to the Theory of Numbers
G. H. Hardy, E. Wright, An Introduction to the Theory of Numbers


Phone: (617) 253-2685

e-mail: mahlburg (at) math (dot) mit (dot) edu.

Office: 2-173 (First floor of Math building, south wing)
Teaching Assistant:

Phone: (617) 452-1199

e-mail: sawyer (at) math (dot) mit (dot) edu.

Office: 2-094 (Basement)

Problem Sets

There will be approximately 10 problem sets, due on Tuesdays in class.  The majority of the problems will be taken from Niven, Zuckerman, and Montgomery.  Please start working on them early, since important results and/or proofs will often appear!

Lecture Schedule

Watch for continued updates throughout the semester.

Sep. 4 (R)
Intro; Pythagorean Numbers I
Niven 5.3, 1.2
Sep.  9 (T) Pythagorean Numbers II; Division algorithm;
gcd's & Euclidean algorithm
Niven 1.2, 1.3
Sep. 11 (R)
More gcd's; lcm's; linear equations; primes
Niven 1.3
Sep. 16 (T)
Fundamental theorem of arithmetic; factorization;  congruences
Niven 1.3, 2.1, 2.2
Sep. 18 (R)
Solving linear congruence equations;  residue systems; Fermat's little theorem
Niven 2.1, 2.2
Sep. 23 (T)
Chinese remainder theorem; Euler's totient function
Niven 2.3
Sep. 25 (R)
Arithmetic functions
Niven 4.2
Sep. 30 (T)
Mobius inversion; Computational techniques; primality testing Niven 4.3, 2.4
Oct. 2 (R)
Primality testing
Niven 2.4
Oct. 7 (T)
RSA cryptography; Multiplicative structure modulo primes
Niven 2.5, 2.8
Oct. 9 (R)
Midterm Exam I

Oct. 14 (T)
Primitive roots modulo primes and composites; binomial coefficients
Niven 2.8, 1.4
Oct. 16 (R)
Binomial coefficients; primitive roots modulo odd prime powers
Niven 1.4, 2.8
Oct. 21 (T)
Primitive roots modulo powers of 2; solving polynomial congruences modulo prime powers
Niven 2.8, 2.6
Oct. 23 (R)
Quadratic equations modulo primes; quadratic residues and Legendre symbol
Niven 3.1
Oct. 28 (T)
Quadratic reciprocity
Niven 3.2
Oct. 30 (R)
Jacobi symbol; generalized quadratic reciprocity
Niven 3.3
Nov. 4 (T)
Pell's equation; rational continued fractions
Niven 7.1, 7.2
Nov. 6 (R)
Periodic continued fractions; Pell's equation
Niven 7.3, 7.7, 7.8
Nov. 11 (T)
No class - Veteran's Day

Nov. 13 (R)
Midterm Exam II

Nov. 18 (T)
Rational approximations of irrationals
Niven 7.4, 7.5
Nov. 20 (R)
Periodic continued fractions and quadratic irrationals; Pell's equation
Niven 7.6, 7.7, 7.8
Nov. 25 (T)
Farey fractions; rational approximations revisited Niven 6.1, 6.2
Nov. 27 (R)
No class - Thanksgiving

Dec. 2 (T)
Rational approximations; partitions; algebraic numbers
Niven 6.2, 10.1, 10.2, 10.3, 9.2
Dec. 4 (R)
Algebraic integers; unique factorization; Riemann hypothesis
Niven 9.2 - 9.8
Dec. 9 (T)
Ramsey Theory; Van der Waerden's theorem

Final Exam
Thursday, Dec. 18
1:30PM - 4:30PM
Room 56-154



Number Theory is one of the oldest subjects in mathematics, and there are a large number of introductory texts.  Please supplement your reading with any of these:


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