Pré-Publication, Document De Travail Année : 2023

Collatz High Cycles Do Not Exist

Kevin Knight
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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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hal-04261183 , version 1 (26-10-2023)

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  • HAL Id : hal-04261183 , version 1

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Kevin Knight. Collatz High Cycles Do Not Exist. 2023. ⟨hal-04261183⟩
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