Collatz conjecture
Number theory
The conjecture that for every positive integer n, repeatedly applying the rule "halve n if it is even, otherwise replace it with 3n + 1" eventually reaches 1.
What is usually left out
The restriction to positive integers is essential, not tidiness. Over all integers the claim is simply false: −1 → −2 → −1 cycles forever, as does −5 → −14 → −7 → −20 → −10 → −5, and 0 maps to itself. Verification has reached beyond 2⁶⁰ without a counterexample, which again is evidence rather than proof.
Related terms
In the atlas
- Collatz conjecture — Dynamical systems, open