Course Topics

Participants in this six-week summer course attend class for 8 hours per week (5 in lecture and 3 in problem seminar). Apart from those classes, students structure their own time, with the understanding that they will spend many hours concentrating on the challenging mathematical ideas presented in class and on the problem sets.

After solving a problem, students are asked to write clear and accurate proofs of all their assertions. This practice with logical thinking and experience in mathematical writing will be of great use in abstract science and mathematics courses taken in college.

Number Theory was chosen as the central topic for the Ross Program because many of its ideas are close to the surface and easily noticed, but deeper concepts and connections are available for exploration. Ross participants investigate this one subject deeply and at length. Although students might not encounter these topics in standard high school or college courses, there is tremendous value in the experience of delving deeply into one subject.

The topics mentioned below appear throughout the problem sets, with different topic-threads often appearing on the same set. By investigating a sequence of related problems over several days, students observe patterns, make conjectures, explore further examples to test the conjectures, formulate theorems, write up proofs, polish the arguments, and investigate generalizations.

Students proceed at various paces through the problem sets. In fact, few students are able to master all the ideas and techniques the first time they see them. The Ross counselors strive to build a “community of scholars” in the dormitory, encouraging everyone to work on these challenging mathematical ideas, to share their ideas and insights, and to experience the joy of solving a difficult problem.

Mathematical Topics:

Euclid’s Algorithm.
Greatest common divisor. Diophantine equation ax + by = c.
Proof of unique factorization in Z.

Modular arithmetic.
Inverses. Solving congruences. Fermat’s Theorem. Chinese Remainder Theorem.
Hensel’s lemma for solving congruences (mod pm).

Binomial coefficients.
Pascal’s triangle. Binomial Theorem.
Arithmetic properties of binomial coefficients, like: (x+y)p = xp + yp (mod p).

Polynomials.
Division algorithm, Remainder Theorem, number of roots.
Polynomials in Zp[x]. Irreducibles and unique factorization.
Z[x] and Gauss’s Lemma.
Cyclotomic polynomials.

Orders of elements.
Units. The group Um. Computing orders.
Cyclicity of Up. For which m is Um cyclic?

Quadratic reciprocity.
Legendre symbols. Euler’s criterion. Gauss’s fourth proof of Reciprocity.
Jacobi symbols.

Continued fractions.
Computing convergents.   |x – p/q| < 1/q2.
Best rational approximations. Pell’s equation.

Arithmetic functions.
phi(n), tau(n), sigma(n), and mu(n). Multiplicative functions.
Sum of   f(d)   as d divides n. Moebius Inversion.
Convolutions of functions.

Gaussian integers:  Z[i].
Norms. Which rational primes have Gaussian factors? Division algorithm.
Unique factorization. Fermat’s two squares theorem.
Counting residues (mod a+bi).

Finite fields.
Characteristic. Frobenius map. Factoring xpn – x.
Counting irreducible polynomials.
Uniqueness Theorem for the field of pn elements.

Resultants.
Discriminant of a polynomial and formal derivatives.
Resultant of two polynomials and relation with Euclid’s algorithm.
Another proof of Quadratic Reciprocity.

Geometry of numbers.
Lattice points. Pick’s Theorem. Minkowski’s Theorem.
Geometric interpretation of the Farey sequence and continued fractions.
Geometric proofs of the two square and four square theorems.

Quadratic number fields.
Which quadratic number rings are Euclidean? For instance
Z[sqrt(d)] is Euclidean when d = -1, -2, 2, 3 but not when d = -3, -5 or 5.
Algebraic integers.