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

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 = 68. 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 respectively.