Slide Index for Slide Show

Structure for Collatz Trajectories
Collatz 3n+1 Conjecture Defined
General Predecessor Tree and Predecessor Formulas
General to Binary Tree Conversion
Features in Binary Predecessor Tree
State Transition Diagram for Predecessor Tree
Abstract Predecessor Tree Features
Features in the Predecessor Trees
Extensions, L.d.a.s Disjoint in Abstract Tree; Binary --> Abstract Mapping
Characterizing Set Elements (dn+c formulas)
Descent Portion of Formula Derivation
Ascent Portion of Formula Derivation
Deriving d of {c[d]} values for L.d.a. elements
Set Densities From Finite L.d.a. Element Sets
Unique Subsets in Abstract Tree
All Integers Appear in the Predecessor Tree
Use Densities to "Count" Integers in Abstract Tree
Densities in a Cardinality Table
Summation of the Densities ==> Integers Covered
Finally, a Proof From Graph Theoretic Properties of the Predecessor Graph Nodes
Graphical Properties of Nodes of Predecessor Trees
Binary Tree Constructed from Graph Elements
Summary of Observations Towards a Proof
Some Slides in Slightly Off-Topic Categories
    Paradoxes Appear and Show Useful Behaviors
       Paradox on (L.d.a. Header Values)/(L.d.a. Leaf Values)
       Paradox on Odd/Even Integer Balance in Predecessor Tree
       Paradox in Feinstein's Proof of Unprovability of Collatz Conjecture
  Alternative Proof Approaches
       L.d.a.s as Sieving Elements
       Finding Any Integer in Abstract Tree
   A Sidelight: Homologous Collatz-Like Iterations
       (3n+3^j)/2^i Homologs of Collatz Conjecture
    What About Infinity?
       Comments on Metrics for Infinity

I just can't bear to throw anything away.
      What Still Remains to Prove?
      A Poorer L.d.a. State Transition Diagram

My Collatz Home Page        Index to Terms Used