The Collatz conjecture

What makes this fascinating

• A rule a child can follow — Take any number: if even, halve it; if odd, triple it and add one. Do you always eventually reach 1?

• Verified to astronomical scale — Checked for every number up to about 2⁶⁸ — all reach 1 — yet no proof covers them all.

• “Mathematics is not yet ready” — Paul Erdős said that about Collatz — and offered a cash prize no one has claimed.

Frequently asked questions

What is the Collatz conjecture?

Take any positive integer: if it is even, halve it; if odd, triple it and add one. The conjecture says you always eventually reach 1, no matter where you start.

Has the Collatz conjecture been proven?

No. It has been verified by computer for numbers up to roughly 2^68, but no proof covers all integers. Paul Erdős remarked that mathematics may not be ready for such problems.

Why is it so hard?

The rule mixes tripling and halving in a way that makes the sequence behave almost randomly, and no known technique controls where it goes in the long run.

More summits in Mathematics

• Riemann Hypothesis

  A 160-year-old pattern in the primes that no one can prove — math's most famous open problem.

• P vs NP

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• Navier–Stokes existence and smoothness

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• Birch and Swinnerton-Dyer conjecture

  A startling claim that a curve's whole-number solutions are encoded in a single function.