MTH 215 Number Theory, Winter semester 2002
- 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
- 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,
- 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
- Niven, Zuckerman and Montgomery, `An introduction to the theory of
numbers', Wiley, fifth edition. [For general reference and exercises]
- Ireland and Rosen, `A classical introduction to modern number theory',
GTM, Springer. [For topics 1-3]
- Apostol, `Introduction to analytic number theory', UTM, Springer.
[For topic 5]
Mid-Sem Test 1
Mid-Sem Test 2