MTH 215 Number Theory, Winter semester 2002


Topics covered

1. The fundamental theorem of arithmetic
Divisibility, Euclidean division, gcd, unique factorization.
2. Congruences
Basic properties, Euler's, Fermat's and Wilson's theorems, linear congruences, Chinese remainder theorem, polynomial congruences and Hensel lifting, roots of polynomials over prime fields.
3. The structure of Z_m^*
Order, primitive roots, nth power residues, Euler's criterion
4. Quadratic residues
Quadratic residues and nonresidues, Legendre symbol, Jacobi symbol, Law of quadratic reciprocity, square roots modulo primes.
5. Number-theoretic functions
Definitions, common arithmetic functions, Dirichlet multiplication and inverse, Möbius inversion formula, multiplicative functions, divisor functions, recurrence functions, combinatorial number theory (Dirichlet's pigeon-hole principle, Inclusion-Exclusion priciple).
6. Some Diophantine equations
Linear equations (ax+by=c), Pythagorean triples (x^2+y^2=z^2), Local and global solutions, Fermat's method of infinite descent, insolvability of x^4+y^4=z^2.
7. Continued fractions
Finite simple continued fractions and representation of rationals, infinite simple continued fractions and representation of irrationals, approximation of irrational numbers by rationals, best approximation.
8. Binary quadratic forms
Indefinite, semidefinite and definite forms, discriminants, representation of integers by forms, equivalence of binary quadratic forms.


Exercise sets

Mid-Sem Test 1
Mid-Sem Test 2
Quiz 1
Quiz 2
Practice exercises
End-Sem Test [Solutions]