A matricial view of the Collatz conjecture. (English) Zbl 1543.15025
The Collatz function \(f\) on \(\mathbb{N}\) is defined by \(f(n)=n/2\) for even \(n\), and \(f(n)=(3n+1)/2\) for odd \(n\). Let \(A=(a_{ij})\) be the infinite matrix with \(a_{ij}=\delta_{f(i),j}\), where \(\delta\) is the Kronecker delta, and let \(A_n\) be its leading \(n\times n\) principal submatrix. For \(n\ge 3\), let \(C_n\) be the \((n-2)\times (n-2)\) principal submatrix of \(A_n\) with entries \(a_{ij}\) for \(3\le i,j\le n\). The author proves that the conjecture stating the nilpotence of any \(C_n\) is equivalent to the Collatz conjecture. This extends the work of J. F. Alves et al. [Linear Algebra Appl. 394, 277–289 (2005; Zbl 1116.11012)] and simplifies a conjecture by D. A. Cardon and B. Tuckfield [Linear Algebra Appl. 435, 2942–2954 (2011; Zbl 1229.15013)].
Reviewer: Jorma K. Merikoski (Tampere)
MSC:
15B34 | Boolean and Hadamard matrices |
11C20 | Matrices, determinants in number theory |
11B83 | Special sequences and polynomials |
40A05 | Convergence and divergence of series and sequences |
References:
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[2] | Cardon, D. A.; Tuckfield, B., The Jordan canonical form for a class of zero-one matrices, Linear Algebra Appl., 435, 11, 2942-2954, 2011 · Zbl 1229.15013 |
[3] | Horn, R. A.; Johnson, C. R., Matrix Analysis, 2013, Cambridge University Press: Cambridge University Press Cambridge · Zbl 1267.15001 |
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