Collatz High Cycles Do Not Exist
Résumé
The Collatz function takes odd $n$ to $(3n + 1)/2$ and even $n$ to $n/2$. Under the iterated Collatz function, every positive integer is conjectured to end up in the trivial cycle 1-2-1. Two types of cycles are of special interest. Consider the set $S$ consisting of the smallest members of all cycles containing the same number of odd terms. The circuit contains the smallest member of $S$, while the high cycle contains the largest. It is known that no circuits of positive integers exist (except 1-2-1); this paper shows that there are likewise no high cycles of positive integers.
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