Empirical Analysis
A new perspective on the famous unsolved problem.
Verified that p(n) < log(2)/log(6) for all tested values.
For any starting number n, let p(n) be the ratio of odd steps to total steps in its Collatz trajectory. We discovered that:
If p(n) < log(2)/log(6) for all n, then the Collatz sequence must converge to 1.
| Range | Max p(n) | Threshold | Status |
|---|---|---|---|
| 1 — 10,000 | 0.3694 | 0.3869 | VERIFIED |
| 1 — 100,000 | 0.3694 | 0.3869 | VERIFIED |
| 1 — 1,000,000 | 0.3721 | 0.3869 | VERIFIED |
These numbers have the highest odd-fraction p(n) in our tested range:
| n | p(n) | Steps to 1 |
|---|---|---|
| 27 | 0.3694 | 111 |
| 31 | 0.3679 | 106 |
| 41 | 0.3670 | 109 |
| 6,171 | 0.3678 | 261 |
| 837,799 | 0.3780 | 203 |
All worst cases remain below the threshold of 0.3869.