A linear algebra approach to the conjecture of Collatz

Alves, João Ferreira; Graça, Mário M.; Maria, Días,; Ramos, J. Sousa

doi:10.1016/j.laa.2004.07.008

Cited by 11 publications

(11 citation statements)

References 1 publication

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the current article we improve the results concerning the existence of this unique orbit. Specifically, we improve results obtained by Sousa Ramos and collaborators in [1]. This is the version of the Collatz function we work with:…”

Section: Introductionsupporting

confidence: 54%

On the orbits associated with the Collatz conjecture

Kauffman¹,

Lopes²

2020

Preprint

View full text Add to dashboard Cite

This article leans on 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. Unfortunately, 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.

show abstract

Section: Introductionsupporting

confidence: 54%

On the orbits associated with the Collatz conjecture

Kauffman¹,

Lopes²

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…The Periodical Trajectory Test examines if there is any periodical trajectory. The test started from number 5×2 60 +1, and step 18 (see [1,7]). Altogether 2 32 cycles were made (i.e.…”

Section: Resultsmentioning

confidence: 99%

“…So far, 1 and 2 are the only known natural numbers whose trajectories are periodical for function T. Let us note that there are also integers whose trajectories are periodical for function T, such as 0, −1, −5, −17. Therefore, an interesting question is whether 1 and 2 are the only natural numbers whose trajectories are periodical for function T. As for this question, there are interesting results from paper [1]. Let us form n × n zeroone matrix A n whose elements are…”

Section: The Trajectory Of Number 649 For Function ()mentioning

confidence: 99%

“…2 16 = 65536. The (Bit vector) array of digits is defined as: array[0] contains the number of digits, array [1] contains the binary digit with weight 0, array [2] contains the digit with weight 1, etc. Such representation of big numbers is chosen because it is directly translated into binary representation of a number; consequently, the division by power of 2 comes down to shifting digits (i.e.…”

Section: Basic Transformations and Proposed Algorithmmentioning

confidence: 99%

See 1 more Smart Citation

Computer-Based Validation of 3n+1 Hypothesis for Numbers 3n−1

2019

Teh. vjesn.

View full text Add to dashboard Cite

show abstract

“…The main purpose of this paper is to describe the Jordan canonical form of A n in terms of the graph Γ n . This description is given in Theorem 6. As a motivating example, let f be the function…”

Section: Introductionmentioning

confidence: 99%