Existence of repetitions of digits in the construction of periodic sequences related to 3x+1 problem: Proof of the theorem on the periods in the 3-adic representation of terms of periodic sequences that appear in Collatz problem

2016 ◽  
Author(s):  
Yagub N. Aliyev
Keyword(s):  
Periodic Sequences ◽  
Collatz Problem

scholarly journals Convergence of Generalized Collatz Problem in k-Adic Field

2021 ◽  
Vol 23 (1) ◽  
pp. 33-47
Author(s):  
Yushu Zhu ◽  
◽  
Sensen Chen ◽  
Qing-You Sun
Keyword(s):  
Fixed Point ◽  
Number Field ◽  
Algebraic Systems ◽  
Periodic Sequences ◽  
Period Problem ◽  
Series Transformation ◽  
Collatz Problem

In this article, we define a new series transformation called transformation and probe into its fixed point and periodicity. We extend the number field of the transform period problem to a wider field. Different constraints are imposed on then different periodic columns are formed after finite transformations. We obtain that their periodic sequences are and respectively after derivation. As an application, it can provide a reference for C problems in more complex algebraic systems.

Some results on the Collatz problem

2000 ◽  
Vol 37 (2) ◽  
pp. 145-160 ◽  
Author(s):  
Ştefan Andrei ◽  
Manfred Kudlek ◽  
Radu Ştefan Niculescu
Keyword(s):  
Collatz Problem

scholarly journals Classes of periodic sequences

1958 ◽  
Vol 2 (2) ◽  
pp. 285-302 ◽  
Author(s):  
N. J. Fine
Keyword(s):  
Periodic Sequences

A new class of quaternary generalized cyclotomic sequences of order 2d and length 2pm with high linear complexity

2018 ◽  
Vol 16 (01) ◽  
pp. 1850006 ◽  
Author(s):  
Longfei Liu ◽  
Xiaoyuan Yang ◽  
Bin Wei ◽  
Liqiang Wu
Keyword(s):  
Finite Fields ◽  
Linear Complexity ◽  
Periodic Sequences ◽  
New Class ◽  
New Family ◽  
Generalized Cyclotomic Classes ◽  
Cyclotomic Sequences

Periodic sequences over finite fields, constructed by classical cyclotomic classes and generalized cyclotomic classes, have good pseudo-random properties. The linear complexity of a period sequence plays a fundamental role in the randomness of sequences. In this paper, we construct a new family of quaternary generalized cyclotomic sequences with order [Formula: see text] and length [Formula: see text], which generalize the sequences constructed by Ke et al. in 2012. In addition, we determine its linear complexity using cyclotomic theory. The conclusions reveal that these sequences have high linear complexity, which means they can resist linear attacks.

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