SUMMARY
The expanded Collatz sequence, defined by replacing 3n+1 with 3n+2k+1 for k=0,1,2,..., has not been extensively explored in existing literature, despite numerous generalizations of the Collatz conjecture. The original sequence for k=0 leads to the known cycle 4-2-1, while varying k values generate distinct cycles, such as 3-12-6-3 for k=1. The primary questions raised include the uniqueness of cycles for each k and whether these cycles are independent of the initial seed number.
PREREQUISITES
- Understanding of the Collatz conjecture and its implications.
- Familiarity with mathematical sequences and cycles.
- Knowledge of generalizations in number theory, particularly those related to the Syracuse function.
- Basic comprehension of mathematical conjectures and their historical context.
NEXT STEPS
- Research the generalization of the Collatz sequence as explored by John Horton Conway.
- Investigate the properties of the Syracuse function and its relation to the Collatz conjecture.
- Examine existing literature on mathematical sequences to identify previous explorations of expanded forms.
- Watch educational videos on the Collatz conjecture to gain visual insights into its cycles.
USEFUL FOR
Mathematicians, number theorists, and enthusiasts interested in the complexities of the Collatz conjecture and its generalizations, as well as those exploring unique properties of mathematical sequences.