Theoretical and computational bounds for m-cycles of the 3 n + 1 problem

@inproceedings{Simons2004TheoreticalAC,
  title={Theoretical and computational bounds for m-cycles of the 3 n + 1 problem},
  author={John Simons and Benne de Weger},
  year={2004}
}
An m-cycle of the 3n+1-problem is defined as a periodic orbit with m local minima. In this article we derive lower and upper bounds for the cycle length and the elements of (hypothetical) m-cycles. In particular, we prove that there do not exist nontrivial m-cycles for 1 ≤ m ≤ 75. Our proofs are based on transcendental number theory, computational diophantine approximation techniques, and a not straightforward generalization of the approach of Steiner and Simons on 1-cycles and 2-cycles… CONTINUE READING

References

Publications referenced by this paper.
Showing 1-10 of 13 references

Linear Forms in Logarithms of Rational Numbers

  • Yuri Nesterenko
  • in: F. Amoroso and U. Zannier (eds.), Springer…
  • 2003

Collatz cycles with few descents

  • T. Brox
  • Acta Arithmetica
  • 2000

An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers

  • E. M. Matveev
  • Part I: Izvestia: Mathematics
  • 1998

The Dynamical System Generated by the 3n+1 Function

  • G. J. Wirsching
  • Lecture Notes in Mathematics 1681, Springer…
  • 1998
2 Excerpts

Formes linéaires en deux logarithmes et déterminants d’interpolation

  • LMN Michel Laurent, Maurice Mignotte, Yuri Nesterenko
  • J. Number Th
  • 1995

CWI Tract 65

  • B.M.M. de Weger, Algorithms for Diophantine Equations
  • Centre for Mathematics and Computer Science…
  • 1989

Lagarias , ” The 3 x + 1 Problem and its Generalizations ”

  • C Jeffrey
  • An Introduction to the Theory of Numbers ”
  • 1981

On the existence of cycles of given length in integer sequences like xn+1 = xn/2 if xn is even, and xn+1 = 3xn+ 1 otherwise

  • C. Böhm, G. Sontacchi
  • Atti Accad. Naz. Lincei, VIII Ser., Rend., Cl…
  • 1978

Similar Papers

Loading similar papers…