A linear algebra approach to the conjecture of Collatz. (English) Zbl 1116.11012
Summary: We show that a “periodic” version of the so-called conjecture of Collatz can be reformulated in terms of a determinantal identity for certain finite-dimensional matrices \(M_k\), for all \(k \geqslant 2\). Some results on this identity are presented. In particular we prove that if this version of the Collatz’s conjecture is false then there exists a number \(k\) satisfying \(k \equiv 8\) (mod 18) for which the orbit of \(\frac{k}{2}\) is periodic.
MSC:
| 11B83 | Special sequences and polynomials |
| 37E05 | Dynamical systems involving maps of the interval |
| 15A15 | Determinants, permanents, traces, other special matrix functions |
References:
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| [3] | Lang, S., Algebra (1993), Addison-Wesley: Addison-Wesley Reading, MA · Zbl 0848.13001 |
| [4] | Wirsching, G. J., The Dynamical System Generated by the \(3n+1\) Function, (in: Lecture Notes in Mathematics, vol. 1681 (1998), Springer-Verlag) · Zbl 0892.11002 |
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