Holy cow! You're still here! Well, OK, we'll get to the point: Tao takes us to the Collatz Conjecture. Proposed in 1937 by German mathematician Lothar Collatz, the Collatz Conjecture is fairly easy to describe, so here we go.
Take any natural number. There is a rule, or function, which we apply to that number, to get the next number. We then apply that rule iteratively, and see where it takes us. The rule is this: If the natural number that we selected is even, divide it by 2; if odd, multiply by 3 and add 1. Consider,for example, 10. It's even, so the rule says to divide by 2, taking us to 5. Now that’s odd, so we multiply 5 by 3 and then add 1, taking us on 16. Now 16 is even, so we cut it in half to get 8. Even again, so halving gets us 4. Now 4 is even, so we take half, getting 2, which is even, and cuts in half to 1.
Start applying the rule to a number other than 10, and you’ll still inevitably end at 1 … we think. That's the Collatz Conjecture.
The validity of this conjecture has been established for all numbers having less than 19 digits, so that covers whatever number you probably had in mind. But even were computers to check for numbers up to 100 digits, or 1,000, that's not a proof for all natural numbers.
The number of iterations it takes a number to first reach 1 is the “stopping time” for that number. In the table below, we just call it number of “Stops”.
Binary search tutorial: https://www.geeksforgeeks.org/binary-search-in-javascript/
| Argument | Stops | Explanation |
|---|---|---|
| 1 | 0 | 1 |
| 2 | 1 | 1 |
| 3 | 7 | 10 5 16 8 4 2 1 |
| 4 | 2 | 2 1 |
| 5 | 5 | 16 8 4 2 1 |
| 6 | 10 | 3 10 5 16 8 4 2 1 |
| 7 | 16 | 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1 |
| 19 | 20 | 58 29 88 44 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1 |
| 21 | 7 | 64 32 16 8 4 2 1 |
| 23 | 15 | 70 35 106 53 160 80 40 20 10 5 16 8 4 2 1 |
On May 6, 2026, Google's gemini provided the following information concerning recent efforts concerning Collatz's conjecture:
The Collatz Conjecture, also known as the 3n + 1 problem, remains one of the most famous unsolved problems in mathematics. While a full proof remains elusive, there have been significant theoretical and computational advancements in recent years.
1. Major Theoretical Breakthrough: Terence Tao (2019)
The most significant recent theoretical progress was made by Fields Medalist Terence Tao in 2019. Using a probabilistic approach and a technique called long-range logarithmic averaging, Tao proved that "almost all" orbits of the Collatz map eventually reach "almost bounded" values (Tao, 2019).
Key Finding: Tao demonstrated that for a starting value x, the sequence will typically drop below any function f(x) that goes to infinity, no matter how slowly that function grows (e.g., f(x) = log log log log x).
Limitation: This is a "measure-theoretic" result. While it proves the conjecture is true for a set of numbers with density 1, it does not rule out the possibility of a rare set of counterexamples that could still diverge to infinity or enter a separate loop.
2. Computational Verification
As of 2026, distributed computing projects have continued to push the boundaries of brute-force verification.
- Exhaustive Search: All starting integers up to 268 (approximately 2.95 × 1020) have been verified to eventually reach 1.
- Cycle Constraints: Calculations have shown that if any non-trivial cycle (other than 4-2-1) exists, it must have a length of at least 114 billion terms.
3. Structural Observations and Challenges
Researchers continue to explore the "fractal-like" nature of the stopping times (the number of steps it takes for n to reach 1).
Statistical Patterns: Recent analysis of spectral exponents indicates that Collatz sequences are not "random noise" but possess a quantifiable, persistent structure.
Undecidability: Some mathematicians, including John Conway, have suggested that the Collatz conjecture might be undecidable—meaning it may be impossible to prove or disprove within the standard axioms of mathematics (ZFC). This is because the 3n + 1 operation can be used to simulate a universal Turing machine.
4. Summary of Recent Status
| Type of Progress | Status |
|---|---|
| Full Proof | Unsolved. No general proof currently exists. |
| Probabilistic Proof | Advanced. Tao's 2019 work proves it for "almost all" numbers. |
| Verification Range | 268. No counterexamples found to date. |
| Minimum Cycle Length | > 114 billion terms. |
"Mathematics may not be ready for such problems." — Paul Erdős
References
Tao, T. (2019). Almost all orbits of the Collatz map attain almost bounded values. arXiv preprint arXiv:1909.03562. https://doi.org/10.48550/arXiv.1909.03562