DeMath — Decentralized math research infrastructure

Statement

Define

T : \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}

by

T(n) = n/2

if

n

is even and

T(n) = 3n + 1

if

n

is odd. Prove that for every positive integer

n

, the iteration

n, T(n), T(T(n)), \ldots

eventually reaches

1

.

Current frontier

Verified for all n ≤ 2^71 (Barina, J. Supercomputing 2025); Tao (Forum of Math. Pi, 2022) proved almost all orbits attain almost bounded values in the sense of logarithmic density.

When this counts as solved

A rigorous proof that every positive integer

n

reaches

1

under the Collatz iteration, OR an explicit

n

whose orbit verifiably diverges or enters a non-trivial cycle distinct from

\{1, 2, 4\}

. The conjecture is binary — partial-density results, extended verification past

2^{71}

, and new stopping-time bounds don't close it.

Classification

binary