The 3x + 1 Problem and its Generalizations

Jeffrey C. Lagarias

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• Introduction
• The 3x+1 problem.
  • A heuristic argument.
  • Behavior of the stopping time function.
  • What is the relation between the coefficient stopping time and the stopping time?
  • How many elements don't have a finite stopping time?
  • Behavior of the total stopping time function.
  • Are there non-trivial cycles?
  • Do divergent trajectories exist?
  • Connections of the problem to ergodic theory.
• Generalizations of problem.
  • Algorithmic decidability questions.
  • Existence of stopping times for almost all integers.
  • Fractional parts of (3/2)^k.
• Conclusion.
• References
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