1 A Graph Theoretical Approach to the Collatz Problem
Heinz Ebert, September 5, 2019 retired Mathematical Technical Assistant of the Institute for Medical Statistics and Computational Biology at the University of Cologne
Abstract:
Andrei et al. have shown in 2000 that the graph of the Collatz function (cid:1)(cid:2) starting with root 8 after the initial loop is an infinite binary tree (cid:3) (cid:4)(cid:2) (8) . According to their result they gave a reformulated Version of the Collatz conjecture: the vertex set (cid:8)((cid:3) (cid:4)(cid:9) (8)) = ℤ (cid:12) . In this paper an inverse Collatz function (cid:1)(cid:9) with eliminated initial loop is used as generating function of a Collatz graph (cid:13) (cid:4)(cid:9) . 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 the branch values where Collatz sequences meet. The proof that the graph (cid:13) (cid:4)(cid:9) is equal to the infinite binary tree (cid:3) (cid:4)(cid:9) (1) with vertex set (cid:8)((cid:3) (cid:4)(cid:9) (1)) = ℤ (cid:12) completes the paper. Key Words:
MSC-Class:
1. Collatz function and conjecture
Let ℕ be the set of nonnegative integers and ℤ (cid:12) be the positive integers. The Collatz problem relates to the Collatz map (cid:1)(cid:2) : ℤ (cid:12) → ℤ (cid:12) : (cid:1)(cid:2)((cid:22)) = (cid:23) (cid:22)/2 if (cid:22) ≡ 0 ((cid:30)(cid:31) 2), (cid:1)(cid:2)((cid:22)) ∈ ℤ (cid:12) The famous 3x+1 or Collatz conjecture now states that for any (cid:22) ∈ ℤ (cid:12) there exists a (cid:16) ∈ ℕ such that (cid:1)(cid:2) (%) ((cid:22)) = 1 , [ (cid:1)(cid:2) (&) ((cid:22)) = (cid:22) and (cid:1)(cid:2) (%(cid:12)’) ((cid:22)) = (cid:1)(cid:2) ∘ (cid:1)(cid:2) (%) ((cid:22))] . This excludes the existence of other loops than the trivial terminal cycle (4, 2, 1, 4, . . . ) and of any divergent sequences.
2. The Collatz graph and a modified Collatz conjecture
Most papers deal with the dynamics of the Collatz function (cid:1)(cid:2) or modified versions of it while pure graph theoretical aspects have seldom been considered. Some exceptions are Andaloro , Andrei et al. ., Lang and Wirsching . Let * : ℤ (cid:12) → ℤ (cid:12) be an arbitrary function. Generally the Collatz graph of the generating function * is defined by: (cid:13) + ((cid:8) + , , + ) ≔ (cid:23) (cid:8) + = ℤ (cid:12) the set of vertices , + = {〈(cid:22), *((cid:22))〉; (cid:22), *((cid:22)) ∈ (cid:8) + } the set of directed edges. Andrei et al. examined the Collatz Graph (cid:13) (cid:4)(cid:2) ((cid:8) (cid:4)(cid:2) , , (cid:4)(cid:2) ) and showed that a subgraph of (cid:13) (cid:4)(cid:2) with the vertex set (cid:8) (cid:4)(cid:2) ⊆ ℤ (cid:12) − {1, 2, 4} and the value 8 as root is an infinite binary tree ? (cid:4)(cid:2) (8) . According to this result they reformulated the Collatz conjecture to be: The vertex set of the tree (cid:8)(? @(cid:2)AA (8)) = ℤ (cid:12) − {1, 2, 4} . Their conclusions also lead to the obvious fact that every (cid:22) ∈ ℤ (cid:12) − {1, 2, 4} could be the root of a tree B (cid:4)(cid:2) ((cid:22)) . Then they concentrate on infinite chain subtrees which are characterized by values which are divisible by 3. Graphs without these chain subtrees are called pruned Collatz graphs . This approach leads to infinite sets of start numbers whose sequences converge at 1.
3. The inverse Collatz function and conjecture
Let the set
C = {(cid:22)|(cid:22) ≡ 4 ((cid:30)(cid:31) 6)} − {4} ⊂ ℤ (cid:12) , then the inverse Collatz map (cid:1)(cid:9) : ℤ (cid:12) → ℤ (cid:12) is: (cid:1)(cid:9)((cid:22)) = (cid:23) 2(cid:22) if (cid:22) ∈ ℤ (cid:12) , (cid:1)(cid:9)((cid:22)) ≡ 0 ((cid:30)(cid:31) 2) ((cid:22) − 1)/3 if (cid:22) ∈ C, (cid:1)(cid:9)((cid:22)) ≡ 1 ((cid:30)(cid:31) 2). Although the two operations of the Collatz function (cid:1)(cid:2) have the above unique inverses in the def-inition of (cid:1)(cid:9) , the function (cid:1)(cid:9) itself is not unique. This is because C is a proper subset of ℤ (cid:12) . So every F ∈ C always has two descendants and is a branch value. It is obvious that the operation simply continues its current sequence while the operation (F − 1)/3 results in an odd number and starts a complete new sequence. As 4 is no element of the set C we avoid the otherwise inevitable initial loop (G, 2, 4, G,…I,… ) . With this the Collatz conjecture reads now: The vertex set of the Collatz tree (cid:3) (cid:4)(cid:9) (1) is (cid:8)((cid:3) (cid:4)(cid:9) ) = ℤ (cid:12) . einz Ebert Sequences of powers of 2, forests and the Collatz tree
4. The infinite forests J K and J L We will now see what happens if we only apply the operation (cid:22)′ = of the generating function (cid:1)(cid:9) of the Collatz graph (cid:13) (cid:4)(cid:9) to all odd numbers (cid:31) ∈ N (cid:10) 6(cid:22)|(cid:22) ≡ 1 (cid:5)(cid:30)(cid:31) 2(cid:7): as start value.
Lemma 4.1: Let ∈ (cid:20) , a fixed (cid:31) ∈ N , the map g: (cid:11) (cid:12) → (cid:11) (cid:12) : O(cid:5)(cid:31), (cid:7) (cid:10) (cid:31) ⋅ 2 Q , then with → ∞ any Collatz graph (cid:13) S is an infinite tree (cid:3) S (cid:5)(cid:31)(cid:7) . Proof: The set of vertices of (cid:3) S ( o ) is (cid:8)(cid:5)(cid:3) S (cid:7) (cid:10) 6T|T (cid:10) (cid:31) ⋅ 2 Q : and the set of edges is ,(cid:5)(cid:3) S (cid:7) (cid:10)6U|U (cid:10) 〈(cid:31) ⋅ 2 Q , (cid:31) ⋅ 2 Q(cid:12)’ 〉: . □ If we substitute (cid:31) = in O(cid:5)(cid:31), (cid:7) we obtain the function
V(cid:5)(cid:16), (cid:7) (cid:10) (cid:5)2(cid:16) (cid:17) 1(cid:7) ⋅ 2 Q which is a family of (cid:16) sequences of powers of 2 . Theorem 4.1:
Let (cid:16) ∈ (cid:20) and ∈ (cid:20) , then with (cid:16) → ∞ and → ∞ the Collatz graph (cid:13) W gener-ated by V(cid:5)(cid:16), (cid:7) is an infinite forest J W of distinct infinite trees (cid:3) W (cid:5)2(cid:16) (cid:17) 1(cid:7) with the set of ver-tices (cid:8)(cid:5)J W (cid:7) (cid:10) (cid:11) (cid:12) . Proof: According to Lemma 4.1 all (cid:3) W (cid:5)2(cid:16) (cid:17) 1(cid:7) are distinct trees. Thus this set of unconnected trees is the forest J W . The set of edges is ,(cid:5)J W (cid:7) (cid:10) 6U|U (cid:10) (cid:5)2(cid:16) (cid:17) 1(cid:7) ⋅ 2 Q , (cid:5)2(cid:16) (cid:17) 1(cid:7) ⋅ 2 Q(cid:12)’ : . The codomain of V(cid:5)(cid:16), 0(cid:7) is the set N and for X 0 the codomain of
V(cid:5)(cid:16), (cid:7) is the set of even numbers
Y (cid:10) 6(cid:22)|(cid:22) ≡ 0(cid:5)(cid:30)(cid:31) 2(cid:7), (cid:22) X 0: . Because of
N ∪ Y (cid:10) (cid:11) (cid:12) the set of vertices of J W is (cid:8)(cid:5)J W (cid:7) (cid:10) (cid:11) (cid:12) . □ Corollary 4.1: All vertices (cid:31) as roots of the trees (cid:3) W (cid:5)(cid:31)(cid:7) have the outdegree UO (cid:12) (cid:5)(cid:31)(cid:7) (cid:10) 1 and all nodes T ∈ Y have one indegree UO [ (cid:5)T(cid:7) (cid:10) 1 and one outdegree UO (cid:12) (cid:5)T(cid:7) (cid:10) 1 . Now we exclusively apply the second operation (cid:22)′ (cid:10) (cid:5)(cid:22) > 1(cid:7)/3 of (cid:1)(cid:9) to all numbers F ∈ C as start values.
Theorem 4.2: Let (cid:16) ∈ (cid:11) (cid:12) , F ∈ C , the map \ : C → (cid:5)N > 61:(cid:7) : \(cid:5)(cid:16)(cid:7) (cid:10) (cid:5)(cid:5)6(cid:16) (cid:17) 4(cid:7) > 1(cid:7)/3 , then with (cid:16) → ∞ the Collatz graph (cid:13) ] is an infinite forest J ] of distinct trees (cid:3) ] (cid:5)F(cid:7) . Proof: ,(cid:5)(cid:13) ] (cid:7) (cid:10) 6U|U (cid:10) 〈6(cid:16) (cid:17) 4, 2(cid:16) (cid:17) 1〉, (cid:16) X 0: and (cid:8)(cid:5)(cid:13) ] (cid:7) (cid:10) C ∪ (cid:5)N > 61:(cid:7) ⊂ (cid:11) (cid:12) . Since all edges U ∈ ,(cid:5)(cid:13) ] (cid:7) are different each edge U itself is a separate tree (cid:3) ] (cid:5)6(cid:16) (cid:17) 4(cid:7) . This set of infinitely many unconnected trees (cid:3) ] (cid:5)F(cid:7) is the forest J ] . □ Corollary 4.2: All roots
F ∈ C of the trees (cid:3) ] (cid:5)F(cid:7) only have an outdegree UO (cid:12) (cid:5)F(cid:7) (cid:10) (cid:31) ∈ (cid:5)N > 61:(cid:7) only have an indegree UO [ (cid:5)(cid:31)(cid:7) (cid:10) 1 .
5. Consequences of the union of J W and J ] The separate application of the operations of the generating function (cid:1)(cid:9) split the Collatz graph (cid:13) (cid:4)(cid:9) into two different forests. The re-union of J W and J ] changes the sets of edges and the in- and outdegrees of the nodes of both forests. Lemma 5.1: ,(cid:5)J W (cid:7) ∩ ,(cid:5)J ] (cid:7) (cid:10) 60: Proof: ,(cid:5)J W (cid:7) (cid:10) 6U|U (cid:10) 〈(cid:5)2(cid:16) (cid:17) 1(cid:7) ⋅ 2 Q , (cid:5)2(cid:16) (cid:17) 1(cid:7) ⋅ 2 Q(cid:12)’ 〉, (cid:16) _ 0, _ 0: and ,(cid:5)J ] (cid:7) (cid:10) 6U|U (cid:10) 〈6(cid:16) (cid:17) 4, 2(cid:16) (cid:17) 1〉, (cid:16) X 0: , hence ,(cid:5)J W (cid:7) ∩ ,(cid:5)J ] (cid:7) ={0}. □ Theorem 5.1: (cid:13) (cid:4)(cid:9) (cid:10) J W ∪ J ] . Proof:
Because of Lemma 5.1 the union ,(cid:5)J W (cid:7) ∪ ,(cid:5)J ] (cid:7) (cid:10) ,(cid:5)(cid:13) (cid:4)(cid:9) (cid:7) introduces no multiple edges. As (cid:8)(cid:5)J W (cid:7) (cid:10) (cid:11) (cid:12) and (cid:8)(cid:5)J ] (cid:7) ⊂ (cid:11) (cid:12) therefore (cid:8)(cid:5)(cid:13) (cid:4)(cid:9) (cid:7) (cid:10) (cid:8)(cid:5)J W (cid:7) ∪ (cid:8)(cid:5)J ] (cid:7) (cid:10) (cid:11) (cid:12) . □ einz Ebert Sequences of powers of 2, forests and the Collatz tree Theorem 5.2: All nodes
T ∈ (cid:8)(cid:5)(cid:13) (cid:4)(cid:9) (cid:7) only have one indegree UO [ (cid:5)T(cid:7) and a maximum outde-gree Δ (cid:12) (cid:5)F(cid:7) (cid:10) 2 . Proof: Due to Theorem 5.1 we can add the in- and outdegrees of the set of vertices. Obviously the degree of the root does not change. According to Corollary 4.1 and 4.2 all vertices F ∈C then have one indegree and two outdegrees and all nodes
T ∈ (cid:5)N > 61:(cid:7) have one indegree and one outdegree. □
6. Proof that the Collatz conjecture is true
The detour due to splitting the Collatz graph (cid:13) (cid:4)(cid:9) into separate components leads to the fact that the vertex set is equal to (cid:11) (cid:12) . To verify that (cid:13) (cid:4)(cid:9) is an infinite binary tree we have to show that there are no circuits and that it is connected.
Theorem 6.1: The Collatz graph (cid:13) (cid:4)(cid:9) is an infinite binary tree (cid:3) (cid:4)(cid:9) (cid:5)1(cid:7) and (cid:8)(cid:5)(cid:3) (cid:4)(cid:9) (cid:7) (cid:10) (cid:11) (cid:12) . Proof: We assume that (cid:13) (cid:4)(cid:9) up to a level V is a bi-nary tree. The figure to the right shows that this is true for the level . According to corollary 5.1 all vertices
T a 1 only have one indegree so UO [ (cid:5)T(cid:7) (cid:10) 1 . No node on level V can have an outgoing edge to a vertex on the levels from up to and in-cluding V since these already have an incom-ing edge. So all successors of the nodes of the level V could only be arranged on the next higher level V′ (cid:10) V (cid:17) 1 . For every new level V′ the constraints - are valid and so the inductive continuation 4 ap-plies ad infinitum since all nodes obviously have successors. Summary of arguments: Theorem 5.1 says (cid:8)(cid:5)(cid:13) (cid:4)(cid:9) (cid:7) (cid:10) (cid:11) (cid:12) . According to the points 1-4 above there cannot exist any circuit in (cid:13) (cid:4)(cid:9) . If we assume that there is a node
T a 1 which has no edge to a predecessor this is a con-tradiction to the fact that the root is the only vertex which has no predecessor. Therefore (cid:13) (cid:4)(cid:9) is connected. According to theorem 5.2 all nodes T have a maximum outdegree Δ (cid:12) (cid:5)T(cid:7) (cid:10) 2 . Thus (cid:13) (cid:4)(cid:9) is an infinite binary tree (cid:3) (cid:4)(cid:9) with vertex set (cid:11) (cid:12) and therefore the Collatz conjecture is true. □
Furthermore, b ecause of the inverse relationship of the two Collatz functions (cid:1)(cid:9) and (cid:1)(cid:2) expressed as (cid:1)b , all edges ,(cid:5)(cid:3) (cid:4)b (cid:7) can be assumed to be undirected and so the Collatz tree (cid:3) (cid:4)b (cid:5)1(cid:7) is weakly connected.
7. References [1]
Andaloro, Paul: The 3x+1 problem and directed graphs, Fibonacci Quarterly 40; 2002; p.43 [2]
Andrei, S. et al.: Chains in Collatz’s tree; Report 217,1999; Dep. of Informatics; Universität Hamburg; http://edoc.sub.uni-hamburg.de/informatik/volltexte/2009/41/pdf/B_217.pdf
Andrei, S. et al.: Some results on the Collatz problem; Acta Informatica 37; 2000; p.145 [3]
Diestel, R.: Graph Theory (GTM 137) 5 th edition; Springer-Verlag; New York; 2016 [4] Lang, W.: On Collatz’ Words, Sequences and Trees; arXiv:1404.2710v1;10 Apr 2014 [5]
Wirsching, G.: The Dynamical System Generated by the 3n+1 Function; Lecture Notes in Mathematics; Vol. 1681; Springer-Verlag; New York; 1998.
Heinz Ebert, Im Heidgen 3, 53819 Neunkirchen-Seelscheid, March [email protected]