A linear algebra approach to the conjecture of Collatz

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 Mk, for all k ⩾ 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 ≡ 8 (mod 18) for which the orbit of k2 is periodic.