Periodic Points Under Iteration of Sum of Squares or Cubes of Digits in Some Positional Systems

DOI

10.2991/ic3me-15.2015.327How to use a DOI?

Keywords

black-hole number iteration sum of squares of digits sum of cubes of digits fixed point periodic point positional system positive integer; formatting; style; styling; insert (key words)

Abstract

There are different periodic and fixed points under iteration of sum of squares or cubes of digits of positive integer in different positional systems. On the iteration of sum of squares of digits, in binary and quaternary these points all converge to fixed point 1. There are three fixed points and one 2-circle in ternary, three fixed points and one 3-circle in quinary, one fixed points and one 8-scircle in senary, five fixed points and two 4-circles in septenary, three fixed points and two 2-circles and one 3-circle in octonary, three fixed points and one 2-circle and one 3-circle in nonary, one fixed point and one 6-circle in hexadecimal. On the iteration of sum of cubes of digits, in binary these points all converge to fixed point 1. There are two fixed points and one 4-circle in ternary . There are only nine fixed points in quaternary. There are three points and one 3-circle in quinary, four fixed points and one 5-circle in senary, seven fixed points, four 2-circles, two 3-circles, one 4-circles and one 9-circlein septenary, six fixed points and one 5-circle in octonary, eight fixed points, two 2-circle, one 5-circle and one 11-circle in nonary, 24 fixed point, two 2-circles, one 3-circle, one 6-circle, two 10-circles, one 14-circle and one 15-circle in hexadecimal. The longest circle is found in base-14 and it is a 27-circle.

Copyright

© 2015, the Authors. Published by Atlantis Press.

Open Access

This is an open access article distributed under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

Download article (PDF)

TY  - CONF
AU  - Wenliang Wu
AU  - Xingju Dang
AU  - Yao Zhang
PY  - 2015/08
DA  - 2015/08
TI  - Periodic Points Under Iteration of Sum of Squares or Cubes of Digits in Some Positional Systems
BT  - Proceedings of the 3rd International Conference on Material, Mechanical and Manufacturing Engineering
PB  - Atlantis Press
SP  - 1696
EP  - 1702
SN  - 2352-5401
UR  - https://doi.org/10.2991/ic3me-15.2015.327
DO  - 10.2991/ic3me-15.2015.327
ID  - Wu2015/08
ER  -