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 set of all normalized Kloosterman sums for a given prime p were computed in a particular order and connected by line segments, the resulting graph contained a pronounced embedded epicycloidal locus. In particular, it appeared that for primes p for which the finite field Fp 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 formulated a number of conjectures regarding the dependence of Kloosterman cardioid geometry on the value of p in the corresponding Kloosterman sums, based on observations arising from the use of software tools developed in related thesis projects.
