A Dual-Radix Modular Division Algorithm for Computing Periodic Orbits within Syracuse Dynamical Systems

This article analyzes the periodic orbits of Syracuse dynamical systems in a novel algebraic setting: the commutative ring of graded $n$-adic integers. Within this context, this article introduces a dual-radix modular division algorithm for computing the graded canonical expansions and graded quotients for a certain class of rational expressions that arise from periodic orbits within these dynamical systems. This division algorithm yields two novel methods for testing the integrality of the B\"{o}hm-Sontacchi numbers.Comment: 21 pages; preprint; submitted; notation edits, supplementary text adde

The 3x+1 problem: An annotated bibliography (1963--1999) (sorted by author)

The 3x+ 1 problem concerns iteration of the map on the integers given by T(n) = (3n+1)/2 if n is odd; T(n) = n/2 if n is even. The 3x+1 Conjecture asserts that for every positive integer n > 1 the forward orbit of n under iteration by T includes the integer 1. This paper is an annotated bibliography of work done on the 3x+1 problem and related problems from 1963 through 1999. At present the 3x+1 Conjecture remains unsolved.Comment: 74 pages latex; 197 references, second title change to distinguish from 3x+1 book; first title change indicates abridgment of earlier versions to papers 1999 and earlier ; part II now covers papers 2000 and later, see arxiv:math.NT/0608208; v.11 cutoff date changed from 2000 to 1999, v.13 added Oulipo reference

On Primitive 3-Smooth Partitions of n

A primitive 3-smooth partition of n is a representation of n as the sum of numbers of the form 2 , where no summand divides another. Partial results are obtained in the problem of determining the maximal and average order of the number of such representations. Results are also obtained regarding the size of the terms in such a representation, resolving questions of Erdos and Selfridge. 0