Rotating Binaries

TL;DR

This paper explores the properties of rotating binary numbers, focusing on divisibility conditions related to their length and Hamming weight, and identifies specific cycle patterns where these divisibility properties hold.

Contribution

It introduces new divisibility criteria for rotated binary numbers and analyzes their connection to binary length, Hamming weight, and rotational cycles.

Findings

Divisibility occurs for cycles of the form kn+c

Connection between rotation, length, and Hamming weight established

Conditions for divisibility in binary rotations identified

Abstract

This paper investigates the behaviour of rotating binaries. A rotation by $r$ digits to the left of a binary number $B$ exhibits in particular cases the divisibility $l\mid N_1(B)\cdot r+1$, where $l$ is the bit-length of $B$ and $N_1(B)$ is the Hamming weight of $B$, that is the number of ones in $B$. The integer $r$ is called the left-rotational distance. We investigate the connection between this rotational distance, the length and the Hamming weight of binary numbers. Moreover we follow the question under which circumstances the above mentioned divisibility is true. We have found out and will demonstrate that this divisibility occurs for $kn+c$ cycles.