18.781
Theory of Numbers
An elementary introduction to number theory with no algebraic prerequisites. Primes, congruences, quadratic reciprocity, diophantine equations, irrational numbers, continued fractions, partitions.
Spring 2012
Topics
- Introduction, Diophantine Equations, Divisibility, GCD
- Euclidean Algorithm, Primes
- Binomial Coefficients, Congruences
- Fermat, Euler, Wilson, Linear Congruences
- Linear Congruences, Chinese Remainder Theorem, Algorithms
- Congruences mod Primes, Order, Primitive Roots
- Primitive Roots (Prime Powers), Index Calculus
- Midterm 1
- Quadratic Residues, Quadratic Reciprocity
- Jacobi Symbol, Computation, Zolotareff’s Definition
- Square Roots, Tonelli’s Algorithm, Number of Consecutive Pairs of Squares mod p
- Cyclotomic Polynomials, Primes Congruent to 1 mod n
- Arithmetic Functions
- Mobius Inversion Formula, Zeta Functions
- Linear Recurrences
- Generating Functions
- Midterm 2
- More on Generating Functions, Two Squares Theorem
- Continued Fractions I
- Continued Fractions II: Inequalities
- Periodic Continued Fractions, Quadratic Irrationalities
- Brahmagupta-Pell Equation
- Four Squares Theorem
- Pythagorean Triples, Fermat Descent
- Rational Points on Conics
Readings
- Niven, Ivan, Herbert S. Zuckerman, and Hugh L. Montgomery. An Introduction to the Theory of Numbers . Wiley, 1991. ISBN: 9780471625469.
- Burton, David M. Elementary Number Theory . Allyn and Bacon, 1976. ISBN: 9780205048144.
- Hardy, G.H., and Edward M. Wright. An Introduction to the Theory of Numbers . Oxford University Press, 1960. ISBN: 9780198533108.
- Davenport, Harold, and James H. Davenport. The Higher Arithmetic: An Introduction to the Theory of Numbers . Cambridge University Press, 2008. ISBN: 9780521722360.
Prerequisites & course information
Prereq: None U (Spring) 3-0-9 units
Sources
- Course catalogue ↗2026–2027
- MIT OpenCourseWare ↗Spring 2012
- MIT OCW syllabus ↗
- MIT OCW calendar ↗
- Historical MIT OpenCourseWare offering. Named reading entries and schedule topics are automatically extracted from the linked source pages.