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Walking Cautiously into the Collatz Wilderness:

Algorithmically, Number Theoretically, Randomly

Edward G. BELAGA,with the co-authorship of M.

MIGNOTTE

Université Louis Pasteur de Strasbourg

Fourth Colloquium on Mathematics and Computer Science : Algorithms, Trees, Combinatorics and Probabilities September 18-22, 2006, Institut Élie Cartan, Nancy, France

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Collatz, or 3n+1 Map

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Collatz conjecture, also known as the Collatz problem, the Syracuse problem, Kakutani's problem, Hasse's algorithm, and Ulam's problem states that a T-trajectory starting fromany natural number n passes ultimately by 1 to becomecyclic trajectory

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Computational verification of the Collatz conjecture• September 21, 2004: 2^58 reached.• December 16, 2004: 2·2^58 reached.• February 23, 2005: 3·2^58 reached.• April 8, 2005: 4·2^58 reached.• May 25, 2005: 5·2^58 reached.• August 2, 2005: 6·2^58 reached.• October 26, 2005: 7·2^58 reached.• February 7, 2006: 8·2^58 reached.• March 29, 2006: 9·2^58 reached.• May 26, 2006: 10·2^58 reached. • By two teams of professonals and amateurs :

Eric Roosendaal (Netherlands) and Tomás Oliveira e Silva (Portugel)

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Three main aspects of the Collatz problem

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Three main mathematical theories involved

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Prehistory : a trip to Newfoundland

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Western Brook Pond. Its cliffs rose 650 m above the water and extend 165 m below

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The Map of Gros Mourne National Park

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Six participants of my first colloquium on the Collatz problem,

Gros Mourne National Park, Newfoundland, 1993

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The main reference :

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To what the Collatz problem resembles and why is it important ?

• (1) Lothar Collatz, a student at the Hambourg University, has written the map and the conjecture into his notebook in July 1932.

More than eighty years before him, Francis Guthrie, a student at University College London, conceived the Four Colour Conjecture. The subsequent efforts to prove this conjecture have led to the creation of graph theory.

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• (2) Like Four Color Problem, the Collatz problem is very hard. Thus, Jeff Lagarias writes in his survey :

Paul Erdös commented concerning its intractability :

« Mathematics is not yet ready for such problems.» Despite this doleful pronouncement, study of the problem has not been without reward … »

In our opinion, Erdös’ remark should be instead perceived as a hint at an exceptionally rich mathematical ``harvest'' which we might expect from the sustained efforts to solve this problem : similarly to the Four Color Problem which was the prime reason of the interest to what is now called Graph Theory.

To what the Collatz problem resembles and why is it important ?

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• (3) The 2nd Collatz problem (according to Jeff Lagarias) :

To what the Collatz problem resembles and why is it important ?

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• (4) Such a conceptual discontinuity is a well-known aspect of the theory of Diophantine equations where learns how to solve one by one equations rather innocently looking for an outsider ; one of the coathors of the present work has also coathored (1996) a paper on the diophantine equations

And do not forget the extraordinary fate of the Fermat Last theorem which inspired the creation of algebraic number theory. As André Weil writes in Scientific American: « To some extent, proving Fermat's Theorem is like climbing Everest. »

To what the Collatz problem resembles and why is it important ?

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• (5) David Hilbert made an attempt to bring some uniformity to the world of diophantine discontinuity : his 10th problem called for the search of a universal procedure answering the basic question whether or not a given diophantine equation has rational solutions. Davis, Putnam, Robinson, and Matiyasevich have proved that no such procedure exists.

• In our opinion, it is the presence of unsolvability phenomena in such a formally complete mathematical framework which « explains » the emerging enormous problematic variability and related to it abrupt conceptual discontinuity, novel meaningfullness, startling and ever growing logical complexity of proofs.

• Against such a background, computational experiment becomes a necessary and respected part of investigations, stimulating the development of new powerful, both general and special, problem oriented methods.

To what the Collatz problem resembles and why is it important ?

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• (6) It was John H. Conway (1972) who brought to the « Collatz wilderness » its most natural universal framework :

To what the Collatz problem resembles and why is it important ?

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• (7) Conway has also proved that multiplicative Conway maps are universal computational devices capable of simulating (universal) Turing machines. Thus, any of the questions which interest us in the case of the Collatz problem, if transferred to Conway maps, become equivalent to the halting problem for a universal Turing machine :

• Do iterations of a Conway map rich 1, or for that matter, any other specific integer ?

• Do iterations of a Conway map end in a cycle ?• Do iterations of a Conway map produce a divergent

trajectory ?

To what the Collatz problem resembles and why is it important ?

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The ultimate generalization of the Collatz problem

Matthews-Watt Conjecture (1985) :

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• Conway’s unsolvability theorem implies that it is undecidable in the general case whether a trajectory starting at an integer is ultimately cyclic.

• The First Matthews-Watts conjecture implies that contracting Conway maps are not universal.

• The exceptional set of a universal (and, thus, expanding) Conway map T is nonempty, infinite, and recursively innumerable.

Some dynamic implications of Conway’s unsolvability theorem

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Existence of divergent trajectories : Collatz dynamical system

simulated by a Markov chain

• Jeff Lagarias and K. Soudararajan (2006)QuickTime™ et undécompresseur

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The problem of existence of cyclic trajectories (1)

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The problem of existence of cyclic trajectories (2)

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The problem of existence of cyclic trajectories (3)

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The problem of existence of cyclic trajectories (4)

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• According to the above formulae, a cyclic (3n+1)-trajectory of the length l and oddlength k exists iff the Collatz number (of the first type)

B=Bk,l=2l-3k>0

divides a Collatz number A=a(P)=a(p1,…,pk-1) (of the second type).

• This diophantine criterion is highly nontrivial : there exist pairs (k,l), such the corresponding numbers B and A are very huge and that A could be «slightly» corrected by a very small divisor to be divided by B.

The problem of existence of cyclic trajectories (5)

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The problem of existence of cyclic trajectories (6)

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The problem of existence of cyclic trajectories (7)

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The problem of existence of cyclic trajectories (8)