On the intimate association between even binary palindromic words and the Collatz-Hailstone iterations

Theophanes Raptis

doi:10.59973/ipil.130

Authors

Theophanes Raptis Information Physics Institute, Athens, Greece

DOI:

https://doi.org/10.59973/ipil.130

Keywords:

Collatz conjecture, Hailstone sequences, Binary palindromes, Iterative dynamics

Abstract

The celebrated 3x+1 problem is reformulated via the use of an analytic expression of the trailing zeros sequence resulting in a single branch formula f(x)+1 with a unique fixed point. The resultant formula f(x) is also found to coincide with that of the discrete derivative of the sorted sequence of fixed points of the reflection operator on even binary palindromes of fixed even length \textit{2k} in any interval [0,...,2^2k-1]. A set of equivalent reformulations of the problem are also presented.

References

R. K. Guy (2004) Unsolved problems in number theory, 3rd Ed., Springer.

D. Applegate, J. C. Lagarias The 3x + 1 semigroup, J. Num. Th., 17 (1): 146–159 (2006).

H. Farkas (2005) Variants of the 3 N + 1 problem and multiplicative semigroups, Geometry, Spectral Theory, Groups and Dynamics: Proceedings in Memory of Robert Brooks. Springer.

Conway, John H. (1972) Unpredictable iterations, Proc. 1972 Number Theory Conf., Univ. Colorado, Boulder. pp. 49–52.

J. C. Lagarias THE 3x + 1 PROBLEM: AN OVERVIEW, in ”The Ultimate Challenge: The 3x + 1 Problem”, AMS, (2010) pp. 3–29. Also in ArXiv:2111.02635.

T. Sterin, D. Woods The Collatz process embeds a base conversion algorithm, Proc. 14th International Conference on Reachability Problems 2020. Also in ArXiv:2007.06979.

T. Sterin Binary Expression of Ancestors in the Collatz Graph in Schmitz, S., Potapov, I. (eds) ’Reachability Problems’ (2020),RP 2020.

Lecture Notes in Computer Science, vol 12448, Springer.

M. G. E. da Luz, D. M. G. dos Santos, E. P. Raposo, G. M.Viswanathan Scale-free behavior in hailstone sequences generated by the Collatz map, Phys. Rev. Res. 3, 013073 (2021).

J G Polli, E P Raposo, G M Viswanathan, M G E da Luz Stochastic-like characteristics of arithmetic dynamical systems:the Collatz hailstone sequences, J. Phys. Complex. 5 (2024) 015011.

A. Khrennikov, M. Nilsson (2004) p-adic Deterministic and Ran- dom Dynamics, Kluwer Academic.

https://oeis.org/A000265

https://oeis.org/A007814

N. Balakrishnan, M. Koutras (2001) Runs and Scans with Applications, Wiley, NY.

Ida Mengyi Pu (2006) Fundamental Data Compression, Science Direct.

https://en.wikipedia.org/wiki/Piadic valuation

J. Berstel, A. Lauve, C. Reutenauer, F. Saliola (2008) Combinatorics on Words: Christoffel Words and Repetitions in Words, CRM Monograph Series, V27.

https://https://abacaba.org/

https://oeis.org/A007953

D. J.S. Robinson (2003) An Introduction to Abstract Algebra Walter de Gruyter.

R. Ron (2006) Introduction to Coding Theory Cambridge Univ.Press.

J. Seberry (2017) Orthogonal Designs: Hadamard Matrices, Quadratic Forms and Algebras, Springer.