Open problem, Arithmetic and number theory, Differential equations and dynamical systems, posed 1937

Collatz conjecture

Open

Define $f(n) = n/2$ if $n$ is even and $f(n) = 3n+1$ if $n$ is odd. The conjecture claims that for every positive integer $n$ , repeatedly applying $f$ eventually reaches $1$ .

Posed by, 1937Lothar Collatz

Research frontier as of 2026

As of 2026 the conjecture remains open. Tao's 2019 result shows that, in a precise density sense, almost every orbit becomes small, but it does not rule out a logarithmic-density-zero set of exceptional starting values that diverge to infinity or fall into an undiscovered cycle other than $4\to2\to1$ . Exhaustive computer search has verified the conjecture for every starting value below roughly $2^{71}$ , with no counterexample or new cycle found.

Best known results

Tao (2019): almost all orbits, in the sense of logarithmic density, attain values below any function tending to infinity, e.g. $\log\log\log n$ .
Exhaustive computer verification confirms the conjecture for every starting value up to roughly $2^{71}$ , with no counterexample found.

Tools and where they stop

Tool	Achieved	Where it stops
Probabilistic / ergodic models of the Syracuse map	Shows almost all orbits become small in a precise density-1 sense (Tao 2019)	Cannot rule out a logarithmic-density-zero set of exceptional orbits that diverge or hit an unknown cycle
Exhaustive computer search	Verifies no counterexample below about $2^{71}$	Cannot prove the statement for all integers, since $\mathbb{N}$ is infinite

Open questions

Does every orbit reach 1, with no exceptions at all — not just density 1?
Are there other cycles besides $4\to2\to1$ , or a divergent orbit?

References

Terence Tao (2022). Almost all orbits of the Collatz map attain almost bounded values · arXiv:1909.03562
Jeffrey C. Lagarias (ed.) (2010). The 3x+1 problem: An overview, in The Ultimate Challenge: The 3x+1 Problem
Jeffrey C. Lagarias (1985). The 3x+1 problem and its generalizations