Authors: Laurent Nedelec
This new text on the Collatz/Syracuse problem is a continuation of the document published in February 2025 on viXra (Laurent Nedelec : An algorithmic approach to solving the Collatz/Syracuse problem. viXra 2502.0056). It explains why the probabilities of divergence for Syracuse trajectories are extremely small. The same conclusion is obtained for non-trivial cycles: their existence is nearly impossible. These two results reinforce the conclusion of the previous text—reached through different methods—namely that the Collatz conjecture is, with very high probability, true.In this new work, we first analyze the structure of the alternations between even and odd iterations within Syracuse trajectories. We then show that N* has an equiprobable structure with respect to even and odd iterations in Syracuse trajectories. Next, we examine how this equiprobability within the structure of N* leads trajectories to be globally decreasing. Finally, we address the implications of these results for the questions of divergent trajectories and non-trivial cycles.
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[v1] 2025-12-12 00:41:14
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