Tree-based Arithmetic and Compressed Representations of Giant Numbers

TL;DR

This paper introduces a tree-based data structure for representing and performing arithmetic on giant numbers, achieving compression and efficiency benefits over traditional bitstring methods, with practical applications in prime and Fermat number computations.

Contribution

It presents a novel tree-based representation for natural numbers that enables efficient arithmetic, compression, and succinct encoding of sets and sequences.

Findings

Compresses large numbers like primes and Fermat numbers into small trees

Enables constant-time exponentiation of two

Allows succinct representations of sets and sequences

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.