Why the Collatz conjecture is still unsolved

The Collatz conjecture, proposed by Lothar Collatz in 1937, says that if you repeatedly halve an even number or triple an odd number and add 1, you always end at 1. Nobody has proved it, even though computers have checked every starting number up to 2⁷¹, about 2.36 sextillion.

The rules are simple enough for a child, but the paths they produce are not. Start at 27 and the sequence climbs and falls for 111 steps before it finally reaches 1. Once there it is stuck in the loop 4, 2, 1 forever. Collatz himself was a respected numerical analyst who wrote major books on solving differential equations and eigenvalue problems; today his name survives mostly through this puzzle.

Why isn’t checking every number up to 2⁷¹ a proof?

Because a counterexample could, in principle, first appear far beyond anything tested. It would have to be a number whose sequence climbs forever, or one that falls into some loop other than 4, 2, 1.

The search has still grown fast: by 2019 mathematicians had tested quintillions of numbers, and in 2025 David Bařina of Brno University of Technology raised the limit to 2⁷¹, using thousands of parallel workers on several European supercomputers. He found four new “path records” but no exception.

Has anyone come close to proving the Collatz conjecture?

The biggest step in decades is a partial result: in 2019 Terence Tao proved that almost all starting numbers, 99% or more in a precise statistical sense, eventually fall to values relatively close to 1. That is still not the conjecture, which claims every number reaches 1. The internet is full of amateur “proofs”; none has held up, and the puzzle has resisted proof for almost 90 years.

Specialists are blunt about the risk. Jeffrey Lagarias of the University of Michigan, its unofficial curator for decades, told Quanta Magazine:

This is a really dangerous problem. People become obsessed with it and it really is impossible.