Description
In August 2007 a parallel/distributed computer program designed and coded by a team of SDSU computer science graduate students, after running continuously for more than seven months on a 45-node commodity PC cluster, emitted a single 5-digit integer: a prime number counterexample disproving the Evans Conjecture on Kloosterman sums. In the course of their research, the students discovered another fascinating fact: when the points on the unit circle corresponding to the summands of all normalized Kloosterman sums for a given prime p were computed and connected by line segments in a particular order, the resulting graph contained a pronounced embedded epicycloidal locus. In particular, it appeared that for primes p for which the finite field GF(p) has g=2 as a generator, the locus is a cardioid whose axis of symmetry differs for different p. This thesis project is one of a group of four related thesis projects aimed at developing a suite of software tools to collect, organize, and visualize empirical data about these "Kloosterman cardioids" to aid in the search for a rigorous mathematical explanation for their existence. In particular, this thesis project investigated various computational geometry algorithms for computing the cusp location and axis of symmetry of a Kloosterman cardioid given the sequence of Kloosterman sums from whose graph the cardioidal locus emerged.
