Total Stopping Times of the Collatz Map up to 10⁷: A Computational Test of the Random-Walk Model

Abstract

The Collatz or 3x + 1 conjecture asserts that repeated application of the map  for even n and    for odd n eventually reaches 1 from every positive integer. Although unproven, it is supported by a heuristic random-walk model which predicts that the total stopping time σ(n), the number of iterations needed to reach 1, has mean approximately 6.952 ln n and a standard deviation proportional to √(Inn ). This study computes σ(n) exactly for every integer up to 10⁷ and tests these predictions. Trajectories were computed in C using memoisation and validated against direct iteration and known records. The longest trajectory below 10⁷ begins at 8,400,511 and takes 429 steps of T (685 steps of the standard map), and the highest point reached, 3.02 × 10¹³, occurs in the trajectory of 6,631,675. A least-squares fit of binned mean stopping times against ln n over 10³ to 10⁷ gives a slope of 6.817 (R² = 0.9996), within 2 per cent of the predicted 6.952. The variance of σ(n) grows in proportion to ln n, with a coefficient of 91 to 97 compared with the predicted 101.4. The standardised stopping times are close to Gaussian in the centre, with 68.7 per cent within one standard deviation, but are positively skewed, with skewness about 0.6 and a long right tail. The results support the random-walk model as a first approximation while identifying systematic departures in the variance and the tail.