On the orbits associated with the Collatz conjecture

Louis H. Kauffman, Pedro Lopes

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)143-154
Number of pages12
JournalLinear Algebra and Its Applications
Publication statusPublished - 15 Apr 2021


  • Collatz conjecture
  • Determinants
  • Permutations
  • Recurrence


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