The Collatz tree as a Hilbert hotel: a proof of the 3x + 1 conjecture
Jan Kleinnijenhuis, Alissa M. Kleinnijenhuis, Mustafa G. Aydogan
TThe Collatz tree is a Hilbert hotel:a proof of the 3 x + 1 conjecture Jan Kleinnijenhuis and Alissa M. Kleinnijenhuis
2, 31
Vrije Universiteit Amsterdam, The Network [email protected] University of Oxford, INET, Mathematical [email protected] MIT, Sloan School of [email protected]
October 28, 2020
Abstract
The yet unproven Collatz conjecture maintains that repeatedly connecting even numbers n to n /2, and odd n to 3 n + 1, connects all natural numbers to the Collatz tree with 1 as its root.The Collatz tree proves to be a
Hilbert hotel for uniquely numbered birds. Numbers divisibleby 2 or 3 fly o ff . An infinite binary tree remains with one “upward” and one “rightward” childper number. Next rightward descendants of upward numbers fly o ff , and thereafter generationafter generation of their upward descendants. The Collatz tree is a Hilbert hotel because its birdpopulation remains equally numerous. The density of unique numbers of birds flying o ff comesnevertheless arbitrarily close to 100% of the natural numbers. The latter proves the Collatzconjecture. The Collatz conjuncture maintains that the Collatz function C ( n ) = n/ n is even, but C ( n ) =3 n + 1 if n is odd, reaches 1 for all natural numbers n , after a finite number of iterations. → → → → → → → ... )9 → → → → → → → → → → → → → → → → → → → → → → → → → This one-sentence conjecture became famous after renowned mathematicians realized that itsproof would not be as easy. In the words of Paul Erd˝os (1913-1996): ”hopeless, absolutely hope-less” ( ). Recently, numerical computations verified the conjecture for all numbers below 2 ( ),while Terence Tao proved the conjecture for “almost all” numbers. Nevertheless, Tao reckons that“the full resolution . . . remains well beyond the reach of current methods” ( ).1 a r X i v : . [ m a t h . G M ] O c t f the Collatz conjecture would hold, then the node set N ( T C ) of one Collatz tree T C with asits trivial cyclic root Ω ( T C ) = { , , } would comprise all natural numbers n . No isolated trajectory would exist, neither a divergent trajectory from n to infinity, nor a nontrivial cycle from any n > n ( ). To be proven is that N ( T C ) = N , with N = { , , , . . . } . The originator of theconjecture, Lothar Collatz (1910-1990), hoped that picturing “number theoretic functions f ( n )”with ”arrows from n to f ( n )”, like in Figure 1, would elicit “relations between elementary numbertheory and elementary graph theory” to prove the conjecture ( ). Figure 1: The original Collatz tree
Figure 1 already distinguishes subset S − of uncolored numbers divisible by 2 and/or 3, whichwill fly o ff first, from green upward numbers and red rightward numbers to be defined in lemmas1 and 2. 2 A Hilbert hotel A Hilbert hotel , first introduced by David Hilbert (1862-1943) in a 1924 lecture, is an infinite hotelthat never gets full upon the arrival of new guests because all hotel guests can move to highernumbered rooms ( ). The key to proving the Collatz conjecture is to grasp from Figures 1 , S , or subsets S , fly away, each with an infinite number of uniquely numbered birds. Their places are takenby infinite, equally numerous swarms of their children, separated by larger and larger numericaldistances, indicating that their density diminishes ever further.Our innovation is to erect a binary tree (Figure 2) instead of a Syracuse tree (explained below)after subset S − with numbers divisible by 2 and/or 3 has flown out. This binary tree empowersa hitherto deficient approach in the research literature to find ever less dense subsets S ⊂ N + , suchthat proving the conjecture on S implies it is true on N ( ) [Suppl. C, Related literature]. Infinitelymany successive subsets S , S , S , S , . . . of intermediary numbers, which sit in between remainingnumbers, fly o ff next from ever less dense binary trees T ≥ , T ≥ , T ≥ , T ≥ , . . . (Figure 3). Periodic it-erations of functions (with definitions 2 and 3) govern which numbers fly o ff in which subset. Thisenables density calculations per period mounting up to familiar sums of infinite geometric series.The proof of lemma 4 reveals that the density of all flown subsets S − , S , S , S , S , · · · , S j , · · ·⊂ N comes arbitrarily close to 100% for j → ∞ . For j → ∞ , the ever lesser density of not yet flownswarms S ≥ j = (cid:85) ∞ i = j S i comes therefore arbitrarily close to 0, which is incompatible with the exis-tence of isolated non-trivial cycles and divergent trajectories. Therefore N ( T C ) = N , the node setof the Collatz tree T C comprises all natural numbers N . The uncolored numbers divisible by 2 and/or 3 in S − are intermediaries between descendantsand ancestors in the remaining set of admissible numbers S ≥ from congruence classes { , } ≡ n mod 6. As Figure 1 shows, numbers divisible by both 2 and 3 , which cannot be reached by 3 n + 1themselves, reduce through numbers that are divisible by 3 only (e.g. 12 = 2 · → · →
3) to evennumbers n that are not divisible by 3 (e.g. 3 → p n ancestors that can be reached from S ≥ by the 3 n + 1 operation, and an odd descendant 2 − q n .For example, 10 = 2 · ∈ S − intermediates between 13 = (cid:16) · − (cid:17) / ∈ S ≥ and 5 ∈ S ≥ .It’s not self-evident how to reconnect the remaining numbers in S ≥ to a binary tree T ≥ . The Syracuse tree function T ( n ) that is commonly used in the research literature connects any odd num-ber n to the most nearby odd number. The inverse Syracuse function connects each odd node n toinfinitely many children T − ( n ) = (2 p n − /
3, corresponding to the infinitely many powers p ∈ N + for which T − ( n ) is odd also [Suppl. Figures 3-4]. For example, n = 1 is connected in parallel to1 = (cid:16) − (cid:17) /
3, thus to itself, and moreover to 5 = (cid:16) − (cid:17) /
3, 85 = (cid:16) − (cid:17) /
3, 341 = (cid:16) − (cid:17) / (cid:16) − (cid:17) / , and so on, e.g. to all green numbers to the left in Figure 1. Infinite amounts ofchildren T − ( n ) preclude easy density calculations per finite period . Recently, a measure of logarith-mic densit y applied to the Syracuse tree enabled Tao nevertheless to prove the Collatz conjecturefor ”almost all” numbers ( ).The binary tree T ≥ in Figure 2 [Suppl. Figures 5,7] connects 1 → → → → → . . . in a serial upward orbit, based on three adaptations of the inverse Syracuse function T − .3 Figure 2:
Binary tree T ≥ with root Ω ( T ≥ ) = 1 of upward and rightward arcs, Legend : upward arcs and nodes in green, rightward arcs and nodes in red
First, odd arguments n and odd outputs T − ( n ) are required to be non-divisible by 3 moreover.Next, n ’s infinite o ff spring T − ( n ) is limited to the minimum value of p , thus to its first-born child,labeled as its rightward , red colored, child R ( n ). Third, the first born T − (3 n + 1) child to n ’s even,already flown, parent 3 n + 1 becomes n ’s upward, green colored, child U ( n ).Definitions 1 and 2 below simplify the upward and rightward functions U and R to periodic functions without unknown minimum powers p . Lemmas 1 and 2 specify the disjoint periodic co-domains N U ⊂ S ≥ and N R ⊂ S ≥ to which U and R map their output. From the periodici-ties in lemmas 1 and 2 it’s a small step to Lemma 3, which states their densities ρ (cid:16) N U (cid:17) = 5 / ρ (cid:16) N R (cid:17) = 27 /
96. Lemma 3 does still not imply that these densities are exclusively based onnumbers included in the binary Collatz tree T C , which has to wait until Lemma 4 . The proofs for the periodicities 6 (definition 1), 18 (definition 2) and 96 (lemmas 1 and 2) arevariants on a similar recipe. First, rewrite each assumed congruence class c ≡ n mod d as anequivalent arithmetic progression n = d i + c for i ∈ N with N = { , N } . Next, verify that thedefinition/lemma holds for i = 0. Finally, prove its independence of the choice of i ∈ N . Definition 1. Upward function U : S ≥ (cid:55)→ N U ; U ( n ) = argmin p (2 p (3 n + 1) − / (cid:40) n + 1 if 1 ≡ n mod 616 n + 5 if 5 ≡ n mod 64 eriodicity proof. The two admissible congruence classes ( c ≡ n mod ⊂ S ≥ in definition 1 cor-respond to the arithmetic progressions n = (6 i + c ) ⊂ S ≥ with c ∈ { , } ⊂ S ≥ . For i = 0 weobtain n = c and U ( n ) = U ( c ) = (2 p (3 c + 1) − /
3. For c = 1, p = 1 renders 7 / (cid:60) S ≥ and nexta minimum p = 2 satisfying U ( c ) = (16 − / ∈ S ≥ . Therefore U ( n ) = (cid:16) (3 · c + 1) − (cid:17) / c + 1 in case 1 ≡ c mod 6. For c = 5, p -values of 1 , / ,
21 and 127 / p = 4 satisfying U ( c ) = (256 − / ∈ S ≥ . Therefore U ( n ) = (cid:16) (3 · c + 1) − (cid:17) / c + 5 in case in case 5 ≡ c mod 6.Let’s prove next that U (6 i + c ) ∈ S ≥ is independent of the choice of i ∈ N + . Collecting termswith and without i gives U ( n ) = U (6 i + c ) = (2 p (3(6 i + c ) + 1) − / (cid:16) p +1 i (cid:17) + ((2 p (3 c + 1) − / i -dependent term (cid:16) p +1 i (cid:17) is divisible by 2 and 3, divisibility of U ( n ) by 2 and 3 onlydepends on the i -independent term (2 p (3 c + 1) − /
3, which is the U ( n ) = U ( c ) = (2 p (3 c + 1) − / The rightward function R ( n ) takes on a di ff erent power p depending on the congruence classesmodulo v =18 of its argument n , where v denotes the argument periodicity. The vector p ( a ) ∈ (cid:42) p ( a ) = [2 , , , , ,
1] corresponds to the admissible congruence classes a ∈ (cid:42) a = [1 , , , , , p ( a ) = 1, which is the case for the contracting congruence classes
11 and 17.
Definition 2. Rightward function R : S ≥ (cid:55)→ N R ; R ( n ) = argmin p (2 p n − / = (cid:42) p ( a ) (cid:42) a (2 n − / ≡ n mod n − / ≡ n mod n − / ≡ n mod n − / ≡ n mod n − / ≡ n mod n − / ≡ n mod Periodicity proof.
The six congruence classes a ≡ n mod
18 in definition 2 correspond to thearithmetic progressions n = 18 i + a ⊂ S ≥ with a ∈ (cid:42) a . For i = 0 , we obtain n = a and R ( n ) = R ( a ) =(2 p a − /
3, which enables the verification that (cid:42) p ( a ) = [2 , , , , ,
1] is the vector of minimum pow-ers of p such that R ( a ) ∈ S ≥ . For example, for a = 11 we obtain p = 1, since R (11) = (2 − / ∈ S ≥ . Let’s now split R ( n ) = R (18 i + a ) = (2 p (18 i + a ) − / i -dependent and i -independent terms.This gives (cid:16) p +1 i ) + (2 p a − (cid:17) / . Since the i -dependent term (cid:16) p +1 i (cid:17) is divisible by 2 and 3,divisibility of R ( n ) by 2 and 3 only depends on the i -independent term (2 p a − /
3, which is the R ( n ) = R ( a ) = (2 p a − / (cid:4) .3 Output periodicity upward function Lemmas 1 and 2 below delineate the periodic codomains N U and N R of the upward and rightwardfunctions U and R . Lemma 1 states that the co-domain N U of the upward function U consists of5 out of the 32 admissible congruence classes modulo 96, also known as the multiplicative group( N / N ) × . Lemma 1. Co-domain upward numbers: N U = { n | { , , , , } ≡ n mod 96 } Proof : Let’s consider the arithmetic progression m = (24 i + a ) ∈ S ≥ for i ∈ N of arguments m forthe rightward function n = U ( m ). For i = 0, we obtain m = a ∈ { , , , , , , , } ∈ S ≥ and n = U ( m ) ∈ { , , , , , , , } ⊂ N U . Reducing numbers to congruence classes gives n mod 96 ∈ { , , , , , , , } , thus n mod 96 ∈ { , , , , } .Let’s prove next that this result for i = 0 is independent of the choice i ∈ N by an inductiveproof that U (24 i + a ) mod 96 = U (24 ( i + 1) + a ) mod 96. By definition 1, proof is required for thecongruence class 1 ≡ n mod 6 that (4 (24 i + a ) + 1) mod 96 = (4 (24 ( i + 1) + a ) + 1) mod 96. Sim-plifying both terms further gives (96 i + 4 a + 1) mod 96 = (96 i + 96 + 4 a + 1) mod 96. Leaving outmultiples of 96 on both sides gives (4 a + 1) mod 96 at each side. Furthermore, proof is required forthe congruence class 5 ≡ n mod i + a ) + 5) mod 96 = ( 16 (24( i + 1) + a ) + 5) mod 96.Simplifying both sides gives (4 · i + 16 a + 5) mod 96 = (4 · i + 4 ·
96 + 16 a + 5) mod 96 . Bothsides give (16 a + 5) mod 96 after leaving out terms with 96. (cid:4) Lemma 2 states that the co-domain N R of the rightward function R comprises 27 out of the 32 admissible congruence classes modulo 96. The admissible N U congruence classes 5, 29, 53, 77 and85 (Lemma 1) are excluded. Lemma 2. Co-domain rightward numbers: N R = (cid:40) n | (cid:40) , , , , , , , , , , , , , , , , , , , , , , , , , , (cid:41) ≡ n mod 96 (cid:41) Proof.
The proof of Lemma 2 will be based on Table 1, with in its first two columns as givens thevector (cid:42) a with six admissible argument congruence classes a ≡ n mod
18 for arguments n ∈ S ≥ ,followed by the corresponding vector (cid:42) p of powers of 2 (see definition 2). Its c i column is for lateruse. 6 able 1 : From argument periodicity 18 in n to rightward output periodicity 96 in R ( n ) 𝑛 ∈ 𝑆 ≥0 𝑅 ( 𝑛 ) ∈ ℕ 𝑅 𝑎 ≡ 𝑛 mod 18 𝑝(𝑎) 𝑏(𝑎) ≡ 𝑅(𝑛) mod 𝑚(𝑎) ℎ(𝑎) = 𝕣 (𝑎) ≡ 𝑅 (
18𝑖 + 𝑎 ) mod 96 𝑐 𝑖 ≡ 𝑅(54𝑗 + 18𝑖 + 𝑎) mod , 𝒂 ⃑⃑ 𝒑 ⃑ 𝒃 ⃑ , 𝒎 ⃑⃑⃑ 𝒉 ⃑⃑ ℕ 𝑅 for 𝑖 = 0, 1, 2 𝑻 ⃑⃑ [[1,0,1,0,1,0],
13 ≡ 𝑅(𝑛) mod 48 [1,0,1,0,1,0],
37 ≡ 𝑅(𝑛) mod 96 [1,0,1,0,1,0], [1,0,1,0,1,0],
17 ≡ 𝑅(𝑛) mod 24 [0,1,0,1,0,1],
11 ≡ 𝑅(𝑛) mod 12 [0,1,0,1,0,1]] Let’s rewrite the six argument congruence classes a ≡ n mod 18 as arithmetic progressions n =18 i + a , with i ∈ N . Splitting R ( n ) = (cid:16) (18 i + a )2 p ( a ) − (cid:17) / i -dependent part m ( a ) = 2 p ( a )+1 intrinsic periodicity of the output R ( n ), and an i -independent part b ( a ) =( a p ( a ) − /
3, which represents the congruence class b ( a ) ≡ R ( n ) mod m ( a ). For example, theargument congruence class a = 13 , with power p ( a ) = 2 , gives as rightward output periodicity m ( a ) = 2 p ( a )+1 b ( a ) = (cid:16) · p ( a ) − (cid:17) / ν is also periodic in multiples of ν – the sinefunction, for example, has a periodicity of 2 π, but also of 4 π , 6 π , 8 π . . . . To obtain the sameoutput periodicity for the rightward function as for the upward function (cf. Lemma 1) the in-trinsic output periodicities m ( a ) ∈ (cid:42) m = [24 , , , , ,
12] are transformed to their
Least CommonMultiple rightward output periodicity v = LCM (24 , , , , ,
12) = 96. For each a ∈ (cid:42) a the num-ber of di ff erent arguments n = 18 i + a , indexed by i , that map their rightward output R (18 i + a )onto a single rightward output period ( j − v . . . jv , indexed by j , is given by the heap vector (cid:42) h = v / (cid:42) m = 96 / [24 , , , , ,
12] = [4 , , , , ,
8] (cf. Table 1). For example, for j = 2 theco-domain range is 96 . . . a = 13 , h (13) = 4 successive arguments, indexed by i = 4 . . .
7, map their rightward outputs R (4 ·
18 + 13) = R (85) = 113 , R (5 ·
18 + 13) = R (103) =137 , R (6 ·
18 + 13) = R (121) = 161 and R (7 ·
18 + 13) = R (139) = 185.Column N R in Table 1 renders the rightward codomain N R of 27 congruence classes with acommon periodicity v = 96 as the union of 6 sets of h ( a ) di ff erent rightward congruence classes R (18 i + a ) mod 96. (cid:4) Lemma 3 explicates the output densities of N U and N R , without considering whether outputs aremapped to the Collatz tree or to isolated trajectories [Suppl. Figure 6].7 emma 3. Output densities of U and R : ρ (cid:16) N U (cid:17) = 596 ; ρ (cid:16) N R (cid:17) = 2796 Proof.
By lemmas 1 and 2, N U and N R are disjoint subsets that cover 5, respectively 27, of the 32admissible congruence classes modulo 96. (cid:4) Let’s first illustrate with Figure 2 our notation for inverses and iterates of the upward and right-ward function that is required for density calculations of infinitely many departing swarms ofinfinitely many uniquely numbered birds. Let U − and R − denote the – downward, respectivelyleftward – inverses of U and R . Let U k ( n ) and R k ( n ) denote the k ’th iterate of U , respectively R .Let U − k ( n ) and R − k ( n ) denote the k ’th iterate of their inverses. For example: U − (5) = 1, U (5) =341, R (5) = 17, R − (17) = 5, U − (1109) = 17 . Each orbit has an inverse orbit, e.g. m = U j R k U − l ( n )implies n = U l R − k U − j ( m ), thus U R U − (5461) = 277 implies U R − U − (277) = 5461.The fast way to grasp that the infinite binary tree T ≥ in Figure 2 allows for iterative departures isto ask, while scrutinizing Figure 3 [Suppl. Notebook, Figures 7-10], what happens after successivedepartures of the subsets S , S , S , · · · whose numbers turn red just before flying o ff .8 ≥ with Ω ( T ≥ ) = U (1) = 5, T ≥ with Ω ( T ≥ ) = U (1) = 85, S departed, S (in red) to fly o ff next; S departed, S (in red) to fly o ff next;Green upward arcs U , red reconnections U R U − Green upward arcs U , red reconnections U R U − T ≥ with Ω ( T ≥ ) = U (1) = 341, T ≥ with Ω ( T ≥ ) = U (1) = 5461, S departed, S (in red) to fly o ff next; S departed, S (in red) to fly o ff next;Green upward arcs U , red reconnections U − RU Green upward arcs U , red reconnections U − RU Figure 3:
Reconnected trees T ≥ , T ≥ , T ≥ , T ≥ after the departures of subsets S , S , S , S The answer is that green-colored numbers that are thereby disconnected are reconnected to eachother in red-colored orbits, indicating that they themselves are ready to fly o ff from the less dense binary trees T ≥ , T ≥ , T ≥ , . . . The top-left pane of Figure 4 shows the less dense tree T ≥ after thedeparture of subset S with rightward orbits of upward descent.In the top-left tree T ≥ generation S of reconnected (red) upward children of S numbers (e.g.the orbit 53 → → → · · · ⊂ S with argument 85 ∈ S > ) replaces numbers on the nodesthat were held in tree T ≥ by subset S (e.g. by the rightward orbit 13 → → → · · · ⊂ S with argument 5 ∈ S > ). In the top-right tree T ≥ generation S with reconnected (red) upwardgrandchildren of S numbers (e.g. the orbit 853 → → · · · ⊂ S with argument 341 ∈ S > ) takesover, and so on. In tree T ≥ j upward generation S j takes over, consisting of bridging, reconnected9rbits B kj ( n ) ⊂ S j with arguments n ∈ S >j as specified in definition 3. Definition 3: Reconnection orbit in node subset S j of tree T ≥ j B kj : S >j (cid:55)→ S j : ∀ k ∈ N : B kj ( n ) = U j R k U − j ( n )Definition 3 implies that subset S j , which will fly o ff next from tree T ≥ j , includes only intermediary numbers in between remaining arguments n ∈ S >j and remaining upward children U >j R k U − j ( n ) ∈ S >j . Therefore numbers on hypothetical isolated trajectories are not included in any subset S j , pro-vided that they are not included in the remaining subsets S >j . Definition 4 specifies the resultingtree T ≥ j . Definition 4: Reconnected tree T ≥ j node set: n ∈ S ≥ j arc set: (cid:40) upward arcs n → U ( n ) , reconnection arcs n → U j R U − j ( n ) (cid:41) cyclic root: Ω (cid:16) T ≥ j (cid:17) = U j (cid:16) Ω ( T ≥ ) (cid:17) = U j (1) = (cid:16) j +1 − (cid:17) / j is odd (cid:16) j +2 − (cid:17) / j is evenFigure 3 shows that the cyclic roots U (1) . . . U (1) of trees T ≥ . . . T ≥ respectively are 5 , , , ,
21 845 ,
349 525 , Ω ( T ≥ ) = U (1) exceeds alreadythe computational record of 2 by September 2020 for the highest number below which theCollatz conjecture is verified ( ). Cyclic roots build the utmost left upward trunk U (1) → U (1) → U (1) → . . . in tree T ≥ (cf. Figure 2). Lemma 4 maintains that the density ρ ( T C ) of bird swarms leaving the Collatz tree approaches1. Its proof outline below rests on the Supplementary Materials with elaborated proofs, plots,examples [Suppl. B11-13] and an underlying Mathematica notebook { Suppl., Notebook].
Lemma 4: density of departed infinite number subsets from the Collatz tree T C ρ ( T C ) = ρ ( S − ) + ρ ( S ) + ρ (cid:0) S , , ,... (cid:1) + ρ (cid:0) S , , ,... (cid:1) = 4 / /
96 + 1 /
21 + 1 /
224 = 1The concepts used in the proofs are already defined and exemplified in the previous subsection.The condensed density proofs below are summarized in Table 15 with in its columns the foursubsets S − , S , S , , ,... and S , , ,... to be considered, and in its rows their densities ρ ( S ) and theperiodicities ν – denoted by Greek nu, and cardinalities per period S | ν ) on which they rest.10 able 2 : Densities, periodicities and cardinalities per period of arguments and outputs Output node subsets 𝑆 −1 𝑆 𝑆 𝑖∈{1,3,5,… } 𝑆 𝑗∈{2,4,6,,… } density 𝜌(𝑆) 𝜌(𝑆 −1 ) = 𝜌(𝑆 ) = 𝜌(𝑆 ) = 𝜌(𝑆 ) = periodicity 𝜈 𝜈 = 𝜈 𝑘 = 𝑣 𝑖𝑘 = 𝑣 𝑗𝑘 = cardinality per period −1 |𝑣) = |𝜈 𝑘 ) = ⋅ 13 𝑘−1 𝑖=1,3,5,⋯ |𝜈 𝑖𝑘 ) = ∙ 13 𝑘−1 𝑗=2,4,6,⋯ |𝜈 𝑗𝑘 ) = ∙ 13 𝑘−1 Argument subsets 𝑆 >0 𝑆 >𝑖 𝑆 >𝑗 periodicity 𝜈 𝜈 𝑘0 = 𝑘+1 𝜈 𝑖𝑘0 = 𝑘+2 𝜈 𝑗𝑘0 = k+2 cardinality per period >0 |𝜈 𝑘0 ) = 𝑘 >𝑖 |𝜈 𝑖𝑘0 ) = 𝑘 >𝑗 |𝜈 𝑗𝑘0 ) = 𝑘+1 Collatz orbits generating function 𝑅 𝑘 (𝑛 ∈ 𝑆 >0 ) ∈ 𝑆 for 𝑘 = 1,2,3, … 𝐵 𝑗𝑘 (𝑛 ∈ 𝑆 >𝑖 ) ∈ 𝑆 𝑖 for odd 𝑖 = 1,3,5, … and 𝑘 = 1,2,3, … 𝐵 𝑗𝑘 (𝑛 ∈ 𝑆 >𝑗 ) ∈ 𝑆 𝑗 for even 𝑗 = 2,4,6, … and 𝑘 = 1,2,3, … Let’s elucidate the structure of this summarizing table, before elaborating in the proofs belowon the formulas in the cells of the table. The bottom of the table shows that for 3 out of the4 output subsets first periodicities and cardinalities per period of their corresponding argumentsubsets have to be determined. Output subset S is generated by iterations k of the rightwardfunction R k ( n ) ∈ S , starting from upward arguments n ∈ S > . The odd upward output subsets S , , ,... are generated by iterations of the reconnection function B kj ( n ), with iterator k for its impliedrightward iterations and iterator i for its implied odd upward iterations, starting from arguments n ∈ S >i . Similarly, the generation of even upward subsets S , , ,... involve an iterator j for evenupward iterations, starting from arguments n ∈ S >j . The table columns show that the Collatz orbit generating function split any node set S ≥ j inarguments S >j that will remain as the node set S ≥ j +1 of tree T ≥ j +1 , after having generated as theircomplement the departing output numbers S j whose density ρ ( S j ) is assessed. The table rowsreveal the periodicities and cardinalities per period that undergird these density assessments. ρ ( S − ) of numbers divisible by and/or Let’s illustrate for the first term, the density ρ ( S − ) of set S − of flown numbers divisible by 2and/or 3, that the density calculations require a periodicity v . The Least Common Multiple of divi-sors 2 and 3 gives as periodicity v = LCM (2 ,
3) = 6. Counting remainders 0 , , ν gives as cardinality per period S − | v ) = 4. Division by ν gives as density ρ ( S − ) = S − | v ) /v = 4 / .2 Density ρ ( S ) of rightward orbits with upward S > arguments The next term is the density ρ ( S ) of rightward function iterates R k ( n ) for k = 1 . . . ∞ , given upwardarguments n ∈ S > , with as a suitable argument periodicity υ = LCM (18 ,
96) = 288 = 2 . Thisperiodicity incorporates the output periodicity 96 of upward arguments in S > (lemma 1) and theinput periodicity 18 of the rightward function (definition 2). The cardinality of upward argumentsper argument period υ = 288 = 3 ·
96 is S ≥ | υ ) = 3 · −→ (cid:16) S ≥ (cid:12)(cid:12)(cid:12) υ (cid:17) = [1 , , , , , { } , { , , , } , { } , { , , , } , { } , { , , , } ]mod 288. The last c i -column of Table 1 shows that to account for all possible congruence classesmodulo 18 at iteration k , an argument periodicity of 3 .
18 = 54 at iteration k − i = 0 , , ff erent output congruence classes R (54 j + 18 i + a ) mod 18 as inputfor iteration k . Consequently, the argument periodicity to account for cardinalities at iteration k amounts to ν k = 3 k − ν = 2 k +1 .At the first rightward iteration starting from upward arguments, the corresponding outputperiodicity ν = υ θ = 1536 = 2 · θ =96 /
18 = 2 − (cf. Table 1) . Correspondingly, the output periodicity at iteration k amounts to ν k = ν k θ k = 2 k +5
3. The multiplier 2 = 16 observed in ν k for successive output periodicities willdetermine the denominator in the density calculation below.Calculating the rightward output cardinality per period requires additionally the heap vec-tor −→ h = [4 , , , , ,
8] from Table 1 to handle intrinsic rightward output periodicities for dif-ferent congruence classes modulo 18 . This gives as output cardinality for the first rightward it-eration given upward arguments −→ (cid:16) S | ν (cid:17) = −→ (cid:16) S > | ν (cid:17) ◦ (cid:42) h = [ , , , , , ◦ [4 , , , , ,
8] =[4 , , , , , k > transformation matrix T from the last column in Table 1 to incorporate that each specific congruence class modulo 18of arguments for the previous iteration k − ff erent congruence classesmodulo 18 of arguments for the current iteration k . Congruence classes 1 , , ,
11 at iteration k − , ,
13 at iteration k . Congruence classes 13 ,
17 at iteration k − , ,
17 at iteration k . Matrix pre-multiplicationwith T (cid:48) , thus with the transpose of T , is required to calculate the cardinality per congruenceclass modulo 18 in iteration k from the cardinality per congruence class modulo 18 in itera-tion k −
1. This gives as cardinalities per congruence class modulo 18 for the second rightwarditerations −→ (cid:16) R ( n ) | ν (cid:17) = (cid:42) T (cid:48) · −→ (cid:16) R ( n ) | ν (cid:17) ◦ (cid:42) h = [180 , , , , , · is used for ma-trix multiplication and ◦ for elementwise multiplication. Iterative application of −→ (cid:18) R k ( n ) | ν k (cid:19) = (cid:42) T (cid:48) · −→ (cid:18) R k − ( n ) | ν k − (cid:19) ◦ (cid:42) h gives as cardinalities per congruence class modulo 18 for the k ’th iteration −→ (cid:18) R k ( n ) | ν k (cid:19) = k − [180 , , , , , −→ (cid:18) R k ( n ) | ν k (cid:19) forsuccessive cardinalities per output period will determine the nominator in the density calculationbelow.The density across iterations k > r = 13 /
16, with a growth rate of 13 in the cardinality (cid:18) R k ( n ) | ν k (cid:19) relative to a growth rate of16 in the output periodicity ν k . This gives ρ ( S ) = (cid:16) R ( n ) | ν (cid:17) /ν + (cid:80) ∞ k =2 (cid:18) R k ( n ) | ν k (cid:19) /ν k = 81 / / /
96. This is equal to the density ρ ( N R ) = 27 /
96 of all rightward numbers (lemma3) . ρ (cid:0) S , , ,... (cid:1) and ρ (cid:0) S , , ,... (cid:1) of upward descendants of S -ancestors Let’s start with a preview. Odd upward subsets ρ (cid:0) S , , ,... (cid:1) and even upward subsets ρ (cid:0) S , , ,... (cid:1) aretreated separately because each second upward iteration invariably yields U ( n ) = 64 n + 21. Thiscan be checked by applying definition 1 twice. It can be seen from Figure 3 in which the odd trees T ≥ and T ≥ to the left are isomorphic with respect to contractions and expansions. The even trees T ≥ (Figure 2), T ≥ and T ≥ (right side Figure 3) are also isomorphic with respect to contractionsand expansions. We will use iterator i to iterate over odd subsets, and iterator j to iterate overeven subsets.Within the bridging functions B ki ( n ) = U i R k U − i ( n ) and B kj ( n ) = U j R k U − j ( n ) The R k iterations obey the S logic above, with increasing powers of 13 in cardinalities at each furtherrightward iteration. The cardinalities of rightward S ancestors apply also their upward childrenin S , upward grandchildren in S , and so on, since each S ancestor has exactly one upward childin S , one upward grandchild S , and so on. Given U ( n ) = 64 n +21, more distant descendants arespread over larger periodicities. With this preview in mind, let’s now clarify the determination ofthe argument periodicities, output periodicities, cardinalities per output period, and densities.Let’s set argument periodicities for odd and even upward generations to ν ik =2 i k +2 re-spectively ν jk =2 j k +2 . This choice guarantees that the reconnection output cardinality in oneoutput period depends only on the argument cardinality within one argument period, and onthe heap vector and the transformation matrix applied to each rightward iteration k in the re-connection function. The corresponding reconnection output periodicities amount to v ki = v ik θ k =2 k +3i + 4 and v jk = v jk θ k = 2 k +3j + 5 . Note that the periodicity, which is the nominator in thedensity calculations below, increases at each further rightward iteration with a factor 2 = 16, butwith a factor 2 · = 64 at each further odd iteration, and also at each further even iteration – inline with U ( n ) = 64 n + 21. For odd generations S i , the cardinality vector of upward arguments n ∈ S >i per argument period ν ik amounts to −→ (cid:16) S >i (cid:12)(cid:12)(cid:12) ν ik (cid:17) = (cid:104) k , k , k , k , k , k (cid:105) . For odd gen-erations, the cardinality vector of upward arguments n ∈ S >i per argument period ν jk amountsto −→ (cid:18) S >j (cid:12)(cid:12)(cid:12)(cid:12) ν jk (cid:19) = (cid:104) k , k , k , k , k , k (cid:105) (cf. Suppl.B12 for lists of argument and output congru-ence classes).We obtain as cardinalities per period (cid:16) S i =1 , , , ··· | ν ki (cid:17) = k − and (cid:18) S j =2 , , , ··· | ν kj (cid:19) = 3 k − ,with increasing powers of 13, just as for the underlying rightward iterations R k ( n ) ∈ S in theprevious subsection – regardless which generation S , S , . . . . is considered, in line with the secondobservation above.Infinite geometric sums result once again when calculating the required densities, with asfinal densities ρ (cid:0) S , , ,... (cid:1) = 1 /
21 and ρ (cid:0) S , , ,... (cid:1) = 1 / ρ (cid:0) S , , ,... (cid:1) = (cid:80) ∞ i =1 , , , ··· (cid:80) ∞ k =1 (cid:16) S i =1 , , , ··· | v ki (cid:17) /v ki is first summated over the inner rightward it-erator k , giving (cid:80) ∞ i =1 , , , ··· (cid:80) ∞ k =1 (2 k − ) / (cid:16) k +3i + 4 (cid:17) . The resulting sum depends only onthe odd upward iterator i , gving (cid:80) ∞ i =1 , , , ··· − i − = 1 /
21. Similarly, the density of even upwardgenerations ρ (cid:0) S , , ,... (cid:1) = (cid:80) ∞ j =2 , , , ··· (cid:80) ∞ k =1 (cid:16) S j =2 , , , ··· | ν kj (cid:17) /ν kj is also summated first over the inner13ightward iterator k , giving (cid:80) ∞ j =2 , , , ··· (cid:80) ∞ k =1 (cid:16) k − (cid:17) / (2 k +3j + 5 ). The resulting sum dependsonly on the even upward iterator j , giving (cid:80) ∞ j =2 , , , ··· − j − = 1 / ρ (cid:0) S , , ,... (cid:1) = 1 /
21 and the even upward densities ρ (cid:0) S , , ,... (cid:1) = 1 /
224 amounts to1 /
21 + 1 /
224 = 5 /
96, which is equal to the density of all upward numbers ρ (cid:16) N U (cid:17) = 5 /
96 (lemma3).
Proving the Collatz conjecture by picturing Collatz trees as a number theoretical Hilbert hotelreveals new links between graph theory and number theory, as anticipated by its originator (5).To our knowledge the Collatz tree is the first Hilbert hotel that is explicitly construed as a numbertree rather than as a number line.A momentous feature of the binary tree is its one-to-one mapping of unique numbers on itsnodes to unique binary sequences of either upward or rightward jumps from its root to each of itsnodes. Just as with prime factorization, natural numbers can be mapped one-to-one to sequences,e.g. 35 =2 ↔ (cid:104) , , , (cid:105) if 35 is factorized, or 35 = 1 −→ → −→ → ↔ (cid:104) , , , (cid:105) if the binary Collatz path from the root to 35 is taken. The proof of the Collatz conjecture mayenhance recently explored applications of the Collatz tree to random number generation, encryp-tion, and watermarking. Acknowledgements.
The authors are grateful to Mustafa Aydogan for comments, to ChristianKoch and Eldar Sultanow for their Python / Github implementation, to Klaas Sikkel for detailedcomments in the run-up to the first arXiv version, and to Wouter van Atteveldt, Sander Dahmen,Cees Elzinga, and Rob de Jeu for various comments on early drafts.14 ncillary Materials.
ElaborationProofsCollatzTree.pdf gives an account of the notation used withfurther examples and elaborations, especially of various tree plots, and of the periodicities and car-dinalities that underlie the density calculations for Lemma 4.
Math12NotebookCollatzTree.nb , withadditionally its static pdf, is the Mathematica 12 version of the Mathematica notebook underlyingthe paper.Works 7-66 from the references below are cited in the ancillary materials. [1] [2] [3] [4] [5] [6] [7][8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29][30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50][51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66]
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