MTH 215 Number Theory, Winter semester 2002

Announcements

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.

References

• 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]

Exercise sets

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