TY - JOUR

T1 - On the orbits associated with the Collatz conjecture

AU - Kauffman, Louis H.

AU - Lopes, Pedro

N1 - Funding Information: Kauffman's work was supported by the Laboratory of Topology and Dynamics, Novosibirsk State University (contract no. 14.Y26.31.0025 with the Ministry of Education and Science of the Russian Federation).Lopes acknowledges support from FCT (Funda??o para a Ci?ncia e a Tecnologia), Portugal, through project FCT PTDC/MAT-PUR/31089/2017, ?Higher Structures and Applications?. Publisher Copyright: © 2021 Elsevier Inc. Copyright: Copyright 2021 Elsevier B.V., All rights reserved.

PY - 2021/4/15

Y1 - 2021/4/15

N2 - This article is based upon previous work by Sousa Ramos and his collaborators. They first prove that the existence of only one orbit associated with the Collatz conjecture is equivalent to the determinant of each matrix of a certain sequence of matrices to have the same value. These matrices are called Collatz matrices. The second step in their work would be to calculate this determinant for each of the Collatz matrices. Having calculated this determinant for the first few terms of the sequence of matrices, their plan was to prove the determinant of the current term equals the determinant of the previous one. They could not prove it for the cases where the dimensions of the matrices are 26+54l or 44+54l, where l is a positive integer. In the current article we improve on these results.

AB - This article is based upon previous work by Sousa Ramos and his collaborators. They first prove that the existence of only one orbit associated with the Collatz conjecture is equivalent to the determinant of each matrix of a certain sequence of matrices to have the same value. These matrices are called Collatz matrices. The second step in their work would be to calculate this determinant for each of the Collatz matrices. Having calculated this determinant for the first few terms of the sequence of matrices, their plan was to prove the determinant of the current term equals the determinant of the previous one. They could not prove it for the cases where the dimensions of the matrices are 26+54l or 44+54l, where l is a positive integer. In the current article we improve on these results.

KW - Collatz conjecture

KW - Determinants

KW - Permutations

KW - Recurrence

UR - http://www.scopus.com/inward/record.url?scp=85099391478&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2020.12.031

DO - 10.1016/j.laa.2020.12.031

M3 - Article

AN - SCOPUS:85099391478

VL - 615

SP - 143

EP - 154

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

ER -