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The Collatz function as an automorphic Cayley colour graph:decidability of an+b conjectures, proof of the 3n + 1 conjecture
Jan Kleinnijenhuis, Alissa M. Kleinnijenhuis, Mustafa G. Aydogan
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The Collatz conjecture states that repeated steps of n→3n+1 at odd numbers and n→n/2 at even numbers amount to walks over root paths to the branching number c=4 in the `trivial' cyclic root 4→2→1→4→… of one connected Collatz graph. The Collatz graph with reverse arrows n →2n and n →(n-1)/3 can be transformed to a 3-regular automorphic Cayley color graph T_(≥0) with as nodes the branching numbers with a remainder of $4$ or $16$ when divided by $18$, building the congruence classes [4,16]₁₈. Labeling the 2ᵏ breadth-first ordered root paths with 2ᵏ binary numbers on the binary number line, for k=1,2,3,…, and pairing them with the 2ᵏ output numbers of these root paths, gives 2ᵏ paired numbers. The 3-regular Cayley graph of these paired branching numbers can be transformed to a 4-regular Middle Pages graph. This 4-regular graph offers to all paired branching numbers from the congruence classes [4,16]₁₈ a unique Eulerian tour to and from the trivial root number pair {0,c=4}. This proves Collatz's 3n+1 conjecture. Whether a specific an+b conjecture offers a Eulerian tour to all its paired branching numbers can be decided by whether it offers such a tour to paired branching numbers lower than 2a³.
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