The Collatz Conjecture

Pick any positive whole number. If it is even, halve it. If it is odd, triple it and add one. Then do it again. And again. The numbers rise like hailstones in an updraft, then fall. Every path anyone has tried lands in the little loop 4, 2, 1. Nobody has proved that every starting number must.

Lothar Collatz started passing the rule around in the 1930s, while he was still a student listening to Edmund Landau and Issai Schur. He never published a grand paper on it. He just kept asking what the iteration does. In 1950 he was still handing the question to people at the International Congress of Mathematicians. The problem picked up other names the way a rumor does: Kakutani’s problem, Ulam’s problem, the Syracuse problem, hailstone numbers, the 3n+1 problem. Same itch under every label.

Paul Erdős put money on the table for progress. Jeffrey Lagarias later recorded him saying mathematics is not yet ready for such problems. The remark travels through those surveys, not through a filmed speech. The meaning is blunt enough either way. A child can run the rule. The proof still refuses to show up.

The numbers really do look like hail. Start at 7 and you climb to 52, then later fall through 16, 8, 4, 2, 1. Start at 27 and the storm lasts 111 steps, with a peak at 9,232. Then that same 4-2-1 trap. The attracting cycle is tiny. The climb can be huge. The conjecture says every positive integer is a hailstone that eventually hits the ground.

Computers have been throwing hailstones for decades. Oliveira e Silva and later Roosendaal pushed the verified range past 10^18, then past 10^20. Barina’s distributed project, published in 2025, checked every start below 2^71. No counterexample appeared. A finite list, even a monstrous one, still leaves infinitely many unchecked integers.

Partial theorems have more texture than the machine search. Terras, in 1976, proved that almost all starting values have a finite stopping time: they eventually go strictly below where they began. Allouche and later Korec pushed those “drop below a power of the start” results further. Krasikov and Lagarias gave density lower bounds on how many starts below a large X are already known to reach 1. In 2019 Tao proved that almost all orbits, measured by logarithmic density, attain values that are almost bounded: smaller than any function that drifts slowly to infinity. None of that is a proof for every n.

Jeffrey Lagarias has been the problem’s long-haul cartographer since the 1980s: surveys, an annotated bibliography, and a collected volume that still orients anyone who wants the literature straight. The honest status line is the same one he has been writing for years. The conjecture looks true. Present methods do not reach a general proof.