abstract
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Optimal bound for the length of rational Collatz cycles

Lorenz Halbeisen and Norbert Hungerbühler

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We consider the arithmetics of Collatz cycles in Q[(2)].
In particular, we prove optimal estimates for the length of a cycle
in terms of its minimum. As an application, we derive an improved
version of Eliahou's criterion, and we show that the length of
(integer) Collatz cycles which do not contain 1, is at least
102,225,496 provided the Collatz conjecture is verified for all
initial values less than or equal to 212,366,032,807,211.