Parallel Integer Relation Detection: Techniques and Applications

Abstract. Let {x1,x2, ·· ·,xn} be a vector of real numbers. An integer relation algorithm is a computational scheme to find the n integers ak, ifthey exist, such that a1x1 + a2x2 + ···+ anxn = 0. In the past few years, integer relation algorithms have been utilized to discover new results in mathematics and physics. Existing programs for this purpose require very large amounts of computer time, due in part to the requirement for multiprecision arithmetic, yet are poorly suited for parallel processing. This paper presents a new integer relation algorithm designed for parallel computer systems, but as a bonus it also gives superior results on single processor systems. Single- and multi-level implementations of this algorithm are described, together with performance results on a parallel computer system. Several applications of these programs are discussed, including some new results in mathematical number theory, quantum field theory and chaos theory. 1