Tree-based Arithmetic and Compressed Representations of Giant Numbers

Abstract: Can we do arithmetic in a completely different way, with a radically different data structure? Could this approach provide practical benefits, like operations on giant numbers while having an average performance similar to traditional bitstring representations? While answering these questions positively, our tree based representation described in this paper comes with a few extra benefits: it compresses giant numbers such that, for instance, the largest known prime number as well as its related perfect number are represented as trees of small sizes. The same also applies to Fermat numbers and important computations like exponentiation of two become constant time operations. At the same time, succinct representations of sparse sets, multisets and sequences become possible through bijections to our tree-represented natural numbers.