Optimal Bounds for the Length of Rational Collatz Cycles
نویسنده
چکیده
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 veriied for all initial values x 0 212 366 032 807 211.
منابع مشابه
More Jabber about the Collatz Conjecture and a Closed Form for Detecting Cycles on Special Subsequences [Assertion: Collatz cycles]
In [ref 1] apparently published in 2006 professor Cadogan presents an interesting function that maps natural numbers into a two dimensional array based on how many ones the number trails off with. The length of trailing one's is said to be the row index. The rest of the number, divided by 2, is the column index. He then shows that applying the Collatz function reduces the ending string of ones ...
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