Math Conjectures

Collatz Conjecture

OPENIconicConjectureProposed c. 1937 · Full conjecture

Canonical statement

Define T:Z>0→Z>0T:\mathbb Z_{>0}\to\mathbb Z_{>0} by
T(n)={n/2,n even,3n+1,n odd. T(n)= \begin{cases} n/2,&n\text{ even},\\ 3n+1,&n\text{ odd}. \end{cases}
For every n∈Z>0n\in\mathbb Z_{>0}, there exists k≥0k\ge0 such that Tk(n)=1T^{k}(n)=1.
View source LaTeX
Define \(T:\mathbb Z_{>0}\to\mathbb Z_{>0}\) by
\[
  T(n)=
  \begin{cases}
    n/2,&n\text{ even},\\
    3n+1,&n\text{ odd}.
  \end{cases}
\] For every \(n\in\mathbb Z_{>0}\), there exists \(k\ge0\) such that \(T^{k}(n)=1\).
NT-DYN-014:[Lagarias1985Collatz][Tao2022Collatz][Lagarias2010CollatzBook]
Number theoryDynamical systems

Notes

Define T(n)=n/2T(n)=n/2 for even nn and T(n)=3n+1T(n)=3n+1 for odd nn. The Collatz conjecture asserts that iterating TT from any positive integer eventually reaches 11. The problem is retrospectively attributed to Collatz around 1937, though surviving accounts do not fix an exact year of proposal; Lagarias's survey traces its circulation and its many equivalent formulations [Lagarias1985Collatz].

Evidence comes in two kinds. Computation has verified the conjecture over vast initial ranges, and the heuristic that trajectories halve more often than they take the 3n+13n+1 step supports typical descent. On the rigorous side, Tao proved that almost all orbits, in the sense of logarithmic density, attain values below any prescribed function tending slowly to infinity [Tao2022Collatz]. The broader literature, including dynamical and logical perspectives on the problem and its generalizations, is collected in the volume edited by Lagarias [Lagarias2010CollatzBook].

None of this excludes a single divergent orbit or an undiscovered nontrivial cycle, and the conjecture remains open; a proof must rule out both possibilities for every starting value.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.