High Performance Computing Meets Experimental Mathematics

In this paper we describe some novel applications of high performance computing in a discipline now known as "experimental mathematics." The paper reviews some recent published work, and then presents some new results that have not yet appeared in the literature. A key technique inovlved in this research is the PSLQ integer relation algorithm (recently named one of ten "algorithms of the century" by Computing in Science and Engineering). This algorithm permits one to recognize a numeric constant in terms of the formula that it satisfies. We present a variant of PSLQ that is well-suited for parallel computation, and give several examples of new mathematical results that we have found using it. Two of these computations were performed on highly parallel computers, since they are not feasible on conventional systems. We also describe a new software package for performing arbitrary precision arithmetic, which is required in this research.