| Summary: | The Collatz conjecture, perhaps the most elementary unsolved problem in mathematics, claims that for all positive integers n, the map n↦n/2 (n even) and n↦3n+1 (n odd) reaches 1 after a finite number of iterations. We examine the Collatz map's orbits, known as hailstone sequences, to determine whether or not they exhibit scale-invariant behavior, in analogy with certain processes observed in real physical systems. We develop an efficient way to generate orbits for extremely large n (e.g., higher than n∼10^{3000}), allowing us to statistically analyze very long sequences. We find strong evidence of a scale-free power law for the Collatz map. We analytically derive the scaling exponents, displaying excellent agreement with the numerical estimations. The scale-free sequences seen in the Collatz dynamics are consistent with geometric Brownian motion with drift, which is compatible with the validity of the Collatz conjecture. Our results lead to another conjecture (conceivably testable through direct, nonetheless very time consuming, numerical simulations): Given an initial n, the average number of iterations needed to reach 1 is proportional, to lowest order, to log[n] (basis 10).
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