New Arithmetic Algorithms for Hereditarily Binary Natural Numbers

Abstract

Hereditarily binary numbers are a tree-based number representation derived from a bijection between natural numbers and iterated applications of two simple functions corresponding to bijective base 2 numbers. This paper describes several new arithmetic algorithms on hereditarily binary numbers that, while within constant factors from their traditional counterparts for their average case behavior, make tractable important computations that are impossible with traditional number representations.