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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