The Conjecture Erdős Said Math Wasn't Ready For

Pick a positive integer n. If it's even, divide by two. If it's odd, multiply by three and add one. Repeat. Take 7: 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1. Take 27: it climbs to 9,232 along the way and bounces around for 111 steps before reaching 1. Lothar Collatz wrote the rule down in 1937. The claim is that every starting number eventually hits 1.

No one has proved it. Computers have checked every integer up to 2^68 — about 295 quintillion — and they all do. But verification is not a proof; the conjecture is an infinite claim about an infinite set, and any one of the integers above the search frontier could in principle escape to infinity or fall into a cycle that doesn't contain 1.

What's strange is the gap between how easy the problem is to state — a child can run the rule — and how thoroughly it resists. Paul Erdős, who was not stingy with conjectures, said "Mathematics may not be ready for such problems" and offered $500 for a solution. Jeffrey Lagarias, who has tracked the literature for decades, called it "an extraordinarily difficult problem, completely out of reach of present day mathematics."

In September 2019 Terence Tao posted a paper proving a version that softens both quantifiers. He showed that for almost all starting integers (in the sense of logarithmic density), the trajectory eventually drops below any function that grows to infinity — a power of log n, log log n, anything. It's the strongest general result on Collatz in decades. It is not the conjecture. The conjecture says "every n" and "reaches 1"; Tao's theorem says "almost every n" and "gets small." The gap between those two statements is where the actual problem still lives.