Collatz Conjecture Stumps Math for 90 Years
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- Lothar Collatz first investigated the eponymous conjecture in the 1930s, proposing that repeatedly halving even numbers or applying 3x+1 to odd ones will always eventually reach 1.
- Paul Erdős declared that "mathematics may not be ready for such problems," reflecting the conjecture's notorious resistance to proof despite its elementary rules.
- Computers have verified the Collatz conjecture for every number up to 2^71, though that still leaves infinitely many numbers unchecked, per the article.
- Jeffery Lagarias documents that the conjecture didn't appear in print until 1971, when it was called "mathematical gossip," before Martin Gardner popularized it in his Scientific American column in 1972.
- Riho Terras proved in 1976 that "almost all" numbers have a finite stopping time, and Ilia Krasikov and Lagarias tightened this in 2002 to show at least x^0.84 numbers below any x will reach 1.
- Terrence Tao delivered the strongest result in 2019, proving almost all numbers can effectively be driven as low as desired — tantalizingly close to a full proof without quite reaching one.
- OpenAI recently used a large language model to find a counterexample solving a different 80-year-old math problem, raising the prospect that AI could similarly crack Collatz.
Why it matters: The Collatz conjecture has resisted proof despite computational verification up to 2^71 and Tao's 2019 result, which drives almost all numbers arbitrarily low but still can't rule out a single counterexample. OpenAI's recent AI counterexample-driven solution to a different 80-year-old problem makes the question of who finishes Collatz first a live one for mathematicians.
