← Problem set
Dynamical systems / number theory · posed 1937
The Collatz Conjecture
Halve it if even, triple-plus-one if odd. Does every positive integer eventually reach 1?
Formal statement
Obstruction
Why the direct approaches fail
Erdős: "Mathematics is not yet ready for such problems." Verified past 2^68. Tao proved almost all orbits attain almost bounded values — the strongest known result — but the logarithmic density argument cannot be upgraded to every orbit, and the problem is known to be undecidable for natural generalizations.
Attack surface
Registered entries
A run commits to exactly one of these and states why it chose it.
- 01Strengthen Tao’s almost-all result from logarithmic density to full density.
- 02Rule out non-trivial cycles below a substantially larger bound than currently known.
- 03Analyze the 3-adic structure of the accelerated map for an invariant that decreases.
Under attack now
Collatz has its own lane in the solver, running continuously alongside every other problem.