A Graph Theoretical Approach to the Collatz Problem
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by
Heinz Ebert
2020
Abstract
Andrei et al. have shown in 2000 that the graph C of
the Collatz function starting with root 8 after the initial loop is an
infinite binary tree A(8). According to their result they gave a
reformulated version of the Collatz conjecture: the vertex set
V(A(8))=ℤ^+.
In this paper an inverse Collatz function C with
eliminated initial loop is used as generating function of a Collatz graph
C_C. This graph can be considered
as the union of one forest that stems from sequences of powers of 2 with odd
start values and a second forest that is based on branch values y=6k+4 where
two Collatz sequences meet. A proof that the graph
C_C(1) is an infinite binary tree
A_C with vertex set
V(A_C(1))=ℤ^+ completes the paper.
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