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Analysis of the Collatz Map: Digital Root Behavior and Loop Impossibility – Adhil Sony


In this paper, we present a novel analysis of the Collatz conjecture. We explore the digital root behavior of numbers under the Collatz map and identify a consistent mapping pattern based on modular arithmetic properties. Furthermore, we introduce a rigorous argument showing that the only possible closed loop in the Collatz map is the trivial loop (4, 2, 1). This result is supported by both analytical reasoning and numerical evidence.

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3 responses to “Analysis of the Collatz Map: Digital Root Behavior and Loop Impossibility – Adhil Sony”

  1. 貓虎皮 Avatar
    貓虎皮

    Why is there a branch containing a mapping from odd number to e2? o1, o2, o0 must be mapped to e1 doesn’t it?

    ( Where “2. Second branch: e1 -> o2 -> e2 -> o1 -> e1” in the third page. )

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    1. Adhil Sony Avatar
      Adhil Sony

      I am really sorry, thank you for pointing out the error in that mapping. It was supposed to be:

      e1-e2-o1-e2-e1

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    2. werewolfdreamily98f9a24440 Avatar
      werewolfdreamily98f9a24440

      The mapping always follows 1-2-1-2-1…

      and for 0, e0-e0-e0 until we reached an odd number with Drmod3 of 0, o0-e1…

      Again, thank you very much for commenting.

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