Lower bounds for the total stopping time of 3𝑥+1 iterates

Abstract

The total stopping time σ ∞ ( n ) \sigma _{\infty }(n) of a positive integer n n is the minimal number of iterates of the 3 x + 1 3x+1 function needed to reach the value 1 1 , and is + ∞ +\infty if no iterate of n n reaches 1 1 . It is shown that there are infinitely many positive integers n n having a finite total stopping time σ ∞ ( n ) \sigma _{\infty }(n) such that σ ∞ ( n ) > 6.14316 log ⁡ n . \sigma _{\infty }(n) > 6.14316 \log n. The proof involves a search of 3 x + 1 3x +1 trees to depth 60, A heuristic argument suggests that for any constant γ > γ B P ≈ 41.677647 \gamma > \gamma _{BP} \approx 41.677647 , a search of all 3 x + 1 3x +1 trees to sufficient depth could produce a proof that there are infinitely many n n such that σ ∞ ( n ) > γ log ⁡ n . \sigma _{\infty }(n)>\gamma \log n. It would require a very large computation to search 3 x + 1 3x + 1 trees to a sufficient depth to produce a proof that the expected behavior of a “random” 3 x + 1 3x +1 iterate, which is γ = 2 log ⁡ 4 / 3 ≈ 6.95212 , \gamma =\frac {2}{\log 4/3} \approx 6.95212, occurs infinitely often.