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An HPC hybrid parallel approach to the experimental analysis of Fermat’s theorem extension to arbitrary dimensions on heterogeneous computer systems

Abstract:

In this work, we consider the empirical, computational, study of the generalized Fermat’s last theorem conjecture recently proposed using Minkowski norms defined within natural vector semispaces. This kind of studies relies on the connection of vector spaces to Boolean hypercubes, which relates to systematic natural number generation and, thus, to the inner structure of natural vector sets. This topic is of interest for the vector description of molecular structures among other uses associated with the development of new Quantitative Structure-Properties Relationships, QSPR. Here, we transform the problem into a combinatorial one susceptible of computational analysis. Due to the computational demands of these studies, we resort to a High-Performance Computing (HPC) approach. Thus, arbitrary precision arithmetic is introduced as well as new algorithmic approaches for the computation of Minkowski norms and the application of Fermat’s condition. Besides, the problem is transformed into an embarrassingly parallel one. This permits to develop a hybrid shared-distributed memory parallel approach. To optimize the efficiency of the process on heterogeneous computer systems, a hierarchical dynamic self-scheduling approach is introduced. We find an increase in efficiency of 23% when using the proposed self-scheduling strategy. Application of this HPC approach to about 6000 billion three-dimensional natural vectors, for Boolean hypercubes up to dimension 15, and a range of Minkowski norm powers between 3 and 64, strongly suggests conjecturing that Fermat’s last theorem can be generalized from dimension two to dimension three.