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Preprints, Working Papers, ... Year : 2021

Is the Syracuse falling time bounded by 12?

Abstract

Let $T \colon \mathbb{N}\to \mathbb{N}$ denote the $3x+1$ function, where $T(n)=n/2$ if $n$ is even, $T(n)=(3n+1)/2$ if $n$ is odd. As an accelerated version of $T$, we define a \emph{jump} at $n \ge 1$ by $\textrm{jp}(n) = T^{(\ell)}(n)$, where $\ell$ is the number of digits of $n$ in base 2. We present computational and heuristic evidence leading to surprising conjectures. The boldest one, inspired by the study of $2^{\ell}-1$ for $\ell \le 500000$, states that for any $n \ge 2^{500}$, at most four jumps starting from $n$ are needed to fall below $n$, a strong form of the Collatz conjecture.
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Dates and versions

hal-03294829 , version 1 (23-07-2021)
hal-03294829 , version 2 (10-08-2021)
hal-03294829 , version 3 (25-08-2021)
hal-03294829 , version 4 (18-10-2021)

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Shalom Eliahou, Jean Fromentin, Rénald Simonetto. Is the Syracuse falling time bounded by 12?. 2021. ⟨hal-03294829v4⟩
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