Format for Final Projects.

1. Riemann Hypothesis and its relationship to prime numbers Summarize the history of this important (and open) mathematical problem. Explain why it is important.

2. Generating Large Primes. Code up the algorithms mentioned on this page, and use it to generate large primes. Compare it to the actual DSS algorithm which is an NIST standard..

3. Four Color Theorem. Four colors suffice to color any map in such a way that no adjacent regions have the same color. Investigate the history of this problem, its importance, and the techniques used in its solution.

4. Can you Hear the shape of a drum?. Can you reconstruct a drum by complete knowledge of all its frequencies (ie eigenvalues or spectra). The answer is no, even in two dimensions. Summarize the history, mathematical techniques, and proof that it is not possible.