The Problem That Even Terence Tao Couldn't Solve

The Collatz conjecture — the simplest unsolved problem in mathematics. Take any positive integer. If it's even, divide by two. If it's odd, multiply by three and add one. Repeat. Does every number eventually reach 1?
The rule was written down by Lothar Collatz in 1937. Since then, every number tested — up to 2⁶⁸, a twenty-digit number — reaches 1. Not a single counterexample exists. And nobody can prove it.

In 2019, Terence Tao published the strongest result anyone has ever reached: for almost all positive integers, in a precise logarithmic density sense, the sequence eventually falls close to 1. Almost all. Not all. That distinction is where the conjecture still lives.

The problem isn't a lack of effort. It's structural. The two halves of the rule — dividing by two and multiplying by three — operate in different mathematical domains. Every technique that handles one goes silent when confronted with the other. No bridge exists between them.
This is the full story — the rule, the sequences, the hailstone numbers that climb before they fall, Tao's 2019 result, and the possibility that this conjecture may not just be unproven, but unprovable.

#CollatzConjecture #Mathematics #TerrenceTao #NumberTheory #UnsolvedMath

TIMESTAMPS:
0:00 The rule stated — two operations, one condition
0:37 Six and eleven — sequences that fall
1:25 27 — the sequence that climbs to 9,232
2:08 Hailstone sequences
2:58 Verified to 2⁶⁸ — still no proof
3:49 Why computation settles nothing
4:27 The structural problem — two domains, no bridge
5:42 Terence Tao's 2019 result
6:29 Almost all. Not all.
7:00 The Collatz tree — following sequences backward
8:06 What we know
8:31 Could it be unprovable?

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SOURCES
Tao, T. (2019) — Almost all orbits of the Collatz map attain almost bounded values (arXiv): https://arxiv.org/abs/1909.03562
Oliveira e Silva, T. (2023) — Empirical verification of the Collatz conjecture: http://sweet.ua.pt/tos/3x+1.html
Lagarias, J.C. (2010) — The Ultimate Challenge: The 3x+1 Problem (AMS): https://bookstore.ams.org/mbk-78

CC ATTRIBUTIONS
Lothar Collatz portrait: Jacobs, Konrad, CC BY-SA 2.0 DE https://creativecommons.org/licenses/by-sa/2.0/de/deed.en, via Wikimedia Commons
Terence Tao portrait: Gert-Martin Greuel, CC BY-SA 2.0 DE https://creativecommons.org/licenses/by-sa/2.0/de/deed.en, via Wikimedia Commons

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