Effective Polynomial Upper Bounds to Perigees and Numbers of (3x+d)-Cycles of a Given Oddlength.

Abstract

For any positive odd integer d not divisible by 3, the arithmetical function Td(m) = { 3m+d 2 , if m is odd m 2 , otherwise generates on the set N of natural numbers a dynamical system Dd. The 3x + d hypothesis, generalizing the well-known 3x + 1 conjecture, asserts that Dd has a finite number of cycles and no divergent trajectories. We study here the cyclic structure of the system Dd, and prove in particular an effective and sharp polynomial upper bound to the number of cycles in Dd with a given number of odd members. 1991 Mathematical Subject Classification : Primary, 11K31, 11K38, 11K55; Secondary, 11B85.

Cite this paper

@inproceedings{Belaga2016EffectivePU, title={Effective Polynomial Upper Bounds to Perigees and Numbers of (3x+d)-Cycles of a Given Oddlength.}, author={Edward G. Belaga}, year={2016} }