The Collatz conjecture is a rule a child can follow, yet it has defeated the greatest mathematicians for ninety years. Computers have verified it for numbers up to 2^68 and it has never failed — but 'never failed so far' is not a proof, and one legendary mathematician called it so seductively dangerous that people were warned not to work on it. In 2019 Terence Tao got closer than anyone in decades, and it still wasn't enough.
Veritasium's definitive intro: the hailstone numbers that soar to thousands before collapsing, the mesmerizing branching tree, and why '3x+1' is a black hole no integer has ever escaped.
The formal statement, its many names (3n+1, the Ulam conjecture, the Syracuse problem), the verification up to 2^68, and every partial result attacked over ninety years.
Professor David Eisenbud on why a problem this trivially simple is 'completely hopeless' — and the famous warning that it was cooked up to distract Cold War mathematicians.
Quanta on what makes Collatz uniquely treacherous — it mixes multiplication and division in a way that scrambles the number-theoretic tools that crack everything else.
How Terence Tao — tipped off by an anonymous commenter on his own blog — proved the conjecture is true for 'almost all' numbers, the biggest leap in decades.
Straight from the Fields medalist's blog. Even if the measure theory runs deep, the framing shows exactly where 'almost all' stops and the unproven abyss of 'all' begins.
The other front: reframing Collatz as a problem for automated SAT-solvers and termination proofs — and why even the machines keep hitting the same wall.
48 pages of the closest humanity has come. For the truly deep dive — the Syracuse map, logarithmic density, and the exact shape of what remains unproven.