Abstract
The Collatz sequence (n 0,n 1,...) is defined by n k+1 = 3n k + 1 or n k /2 depending on n k being odd or even, respectively. The Collatz conjecture (one of the most challenging open problems in Number Theory) states then that n k = 1 for some k depending on n 0. This conjecture can be reformulated in a variety of ways, some of them seemingly more amenable to the methods of discrete mathematics. In this paper, we derive one such equivalent formulation involving exponential Diophantine equations. It follows that if the Collatz conjecture is true, then any number can be represented as sums of positive powers of 2 and negative powers of 3.
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Amigó, J.M.: Accelerated Collatz dynamics. Trabajos I+D I-2000-5, Operations Research Center of Miguel Hernandez University (available from www.crm.es/Publications/Preprints01.htm, preprint 474) (2000)
Andrei S., Masalagiu C. (1998): About the Collatz conjecture. Acta Informatica 35, 167–179
Andrei S., Kudlek M., Niculescu R.S. (2000): Some results on the Collatz problem. Acta Inform. 37(2): 145–160
Böhm, C., Sonntacchi, G.: On the existence of cycles of a given length in integer sequences like x n+1 = x n /2 if x n even, and x n+1 = 3x n + 1 otherwise, Atti. Accad. Naz. Lincei, VIII Ser., Rend., Cl. Sci. Fis. Mat. Nat. LXIV, 260–264 (1978)
Chamberland, M.: An update on the 3x + 1 problem. (available from www.math.grin.edu/~ chamberl/papers.html)
Lagarias J.C. (1985): The 3x + 1 problem and its generalizations. Am. Math. Month. 88, 3–23
Letherman S., Schleicher D., Wood R. (1999): The 3n+1-problem and holomorphic dynamics. Exp. Math. 8, 241–251
Schroeder M.R. (1984): Number Theory in Science and Communication. Springer, Berlin Heidelberg New York
Wagon S. (1985): The Collatz Problem. The Math. Intelligencer 7, 72–76
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This work has been supported by the Spanish Ministry of Science and Education, grant MTM2005-04948
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Amigó, J.M. Representing the integers with powers of 2 and 3. Acta Informatica 43, 293–306 (2006). https://doi.org/10.1007/s00236-006-0021-0
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DOI: https://doi.org/10.1007/s00236-006-0021-0