Is the Collatz conjecture true?

The question

Take any positive whole number. If it is even, halve it. If it is odd, multiply it by 3 and add 1. Repeat.

Starting from 6, you get 3, 10, 5, 16, 8, 4, 2, 1. Starting from 27, the numbers climb as high as 9,232 and wander for 111 steps before they too fall to 1. Once you reach 1, you are stuck in the loop 1, 4, 2, 1.

The Collatz conjecture says that every starting number eventually reaches 1. The German mathematician Lothar Collatz is said to have considered the problem in the 1930s; it has also been known as the 3n + 1 problem, the Syracuse problem and Ulam's problem. A child can check it with pencil and paper. No mathematician has been able to prove it.

What we know

It has been checked by computer for every starting number up to 2⁷¹, about 2.4 × 10²¹. All of them reach 1.
If the conjecture failed, there would have to be either a number whose sequence grows forever, or a different loop. Any such loop has been shown to be enormously long — hundreds of billions of steps at least.
In 1976 Riho Terras showed that for almost all starting numbers, the sequence eventually drops below where it began. That is a necessary first step, but "almost all" leaves room for exceptions, and dropping below the start is not the same as reaching 1.
In 2019 Terence Tao proved a much stronger statement: for almost all starting numbers, the sequence eventually becomes smaller than any slowly growing function of the start — for example, smaller than the logarithm of the logarithm of it. His paper appeared in 2022. It is the strongest result known, and Tao himself noted that it does not come close to settling the conjecture.
Very similar problems are known to be impossible to decide. In 1972 John Conway showed that some generalisations of the Collatz rules can simulate any computer program, so no algorithm can decide where their sequences end.

Main hypotheses

It is true

Almost everyone expects the conjecture to be true. A simple probabilistic argument supports it: after each odd step the number is multiplied by about 3, but on average it is then halved about twice, so in the long run it shrinks by a factor of about 3/4 per odd step. Sequences should therefore drift downward and eventually hit 1.

It is true but unprovable

Because generalised versions are undecidable, some mathematicians suspect that the Collatz conjecture itself may be true but impossible to prove from the standard axioms. There is no evidence for this beyond the analogy.

A counterexample exists but is unimaginably large

A minority consider it possible that a huge exception exists, far beyond what computers can reach. Nothing supports this, but nothing rules it out.

Where understanding ends

The probabilistic argument works only on average. The difficulty is that a specific number's sequence is completely determined, not random, and no one knows how to show that a deterministic sequence actually behaves like the average. The tools that give "almost all numbers" results cannot reach "every number".

The problem mixes two kinds of arithmetic that don't normally talk to each other: multiplication by 3 and division by 2. Their interaction scrambles the digits of a number in a way that looks random, and current mathematics has no general method for controlling it. This is why Paul Erdős reportedly said that mathematics is not yet ready for such problems.

There is also a link to computation. Since closely related rules can encode any program, the Collatz problem sits near the boundary of what can be decided at all — a theme that connects it to questions like P versus NP.

What would settle it

A proof: most likely through a new way of showing that a deterministic process behaves like a random one, which would matter far beyond this problem.
A counterexample: a number that never reaches 1, found by computation or constructed by theory.
A proof that the conjecture is independent of the standard axioms, which would itself be a major result.