Corpus ID: 119734709

On the 3x+1 conjecture

@article{Hellekalek2016OnT3,
  title={On the 3x+1 conjecture},
  author={Peter Hellekalek},
  journal={arXiv: Number Theory},
  year={2016}
}
In this paper, we discuss the well known 3x+1 conjecture in form of the accelerated Collatz function T defined on the positive odd integers. We present a sequence of quotient spaces and an invertible map that are intrinsically related to the behavior of T. This approach allows to express the 3x+1 conjecture in form of equivalent problems, which might be more accessible than the original conjecture. 

References

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On the (3N+1)-conjecture
The 3n + 1 conjecture, also known as the Collatz conjecture, is an unsolved conjecture in number theory. The 3n + 1 function T : N+ → N+ is defined as: T (n) = { n/2 if n ≡ 0 (mod 2) (3n + 1)/2 if nExpand
Strongly sufficient sets and the distribution of arithmetic sequences in the 3x+1 graph
TLDR
This paper constructs sufficient sets of arbitrarily low asymptotic density in the natural numbers and shows that the 3 x + 1 digraph exhibits a surprising and beautiful self-duality modulo 2 n for any n, and proves that it does not have this property for any other modulus. Expand
The Ultimate Challenge: The 3x+1 Problem
TLDR
This book reports on what is known on the problem of the 3x 1 problem, relating it to number theory and dynamical systems, to Markov chains and ergodic theory, and to logic and the theory of computation. Expand
An Introduction to Ergodic Theory, volume 79 of Graduate Texts in Mathematics
  • Author’s address: Peter Hellekalek, Dept. of Mathematics, University of Salzburg, Hellbrunnerstrasse
  • 1982