Conjecture Fieldbook

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Collatz conjecture

Canonical problem dossier

Repeatedly halve even integers and replace odd integers n by 3n + 1. Does every positive starting value eventually reach 1?

Status
Open
Field
Discrete mathematics
Verification
Reviewed dossier
Source grade
secondary
Reviewed
2026-07-27T18:04:33.834Z
Reviewer
Conjecture Fieldbook maintainers

Formal expression

T(n)={n/2n≡0(mod2)3n+1n≡1(mod2)T(n)=\begin{cases}n/2&n\equiv0\pmod2\\3n+1&n\equiv1\pmod2\end{cases}

Why it matters

Its rule is elementary, yet known methods do not control every orbit. It is a compact example of unpredictable behavior arising from simple arithmetic.

Context and known approaches

Large computational ranges have been checked, and strong statistical results are known, but neither a universal proof nor a counterexample is known.

Formal boundary

For T(n)=n/2 when n is even and T(n)=3n+1 when n is odd, every n∈N is conjectured to have an iterate Tᵏ(n)=1.

Known partial results

  • Large finite checks and density results do not establish the universal claim.

Source trail

  1. Wikipedia — Overview, checked ranges, and reference trail.
  2. Wikipedia open-problems list — Composite index by mathematical field.
  3. Wikipedia list subsection — Placement in the accepted revision-pinned living-list snapshot.