## Number Theory Projects

## [latexpage]Undergraduate Projects

These projects were developed as final projects for the Undergraduate Number Theory course at Duke University, which I taught for four consecutive Spring semesters (2004-2007). Students were asked to choose a “serious topic in Number Theory” (of which the projects were my suggestions with useful references) to write a final paper of at least 10 pages explaining their topic (i.e. explaining and proving their main theorem) and also to give a one hour final presentation to their peers (fellow undergraduates in the class). This assignment was supported with suggested/scheduled individual weekly meetings with me of 30 minutes to one hour per student to discuss progress and answer questions.

**Brief Project Descriptions**

**Detailed Project Descriptions and References**

- Primes in Progressions
- The Prime Number Theorem
- Class numbers and binary quadratic forms
- Pell’s equation and the Continued Fraction expansion of $\sqrt D$
- Dirichlet’s Class Number Formula
- Geometry and Sums of four squares: Minkowski’s Theorem
- Arithmetic and Sums of four squares: Quaternions
- Algebra and Sums of four squares: Local-Global Principle and Class Numbers
- Analysis and Sums of four squares: Siegel’s theorem
- Local-global principle for rational points on conics and $p$-adic numbers
- Algebra and Analysis in the $p$-adic numbers
- Public Key Cryptography
- Factoring and Primality testing
- Elliptic Curve Cryptography
- ** The group law on Elliptic curves, and finding rational and integral points

(FIX MISSING REFERENCES!) - Congruence and Hasse-Weil zeta functions
- Solving $x^3+ y^3 = z^3$ and $x^4 + y^4 = z^4$
- Cubic and Biquadratic Reciprocity
- Cyclotomic numbers and Fermat’s Last Theorem
- Special Values of the Riemann Zeta function $\zeta(s)$
- Constructibility of Regular Polygons

## Graduate Projects

These projects were developed as final projects for the Graduate Number Theory course on Modular Forms at the University of Georgia, which I taught in Spring 2009. Students were asked to choose a “serious topic in Number Theory” (of which the projects were my suggestions with useful references) to write a final paper of at least 10 pages explaining their topic (i.e. explaining and proving their main theorem) and also to give a one hour final presentation to their peers (fellow graduate students in the class).

- The Prime Number Theorem
- The Eichler-Shimura Relations
- The Rankin-Selberg Method of “Unfolding”
- The Congruent Number Problem