For every positive integer n∈Z_>0, define the Collatz map T:Z_>0→Z_>0 by T(n)=n/2 if n is even and T(n)=3n+1 if n is odd. The conjecture asserts that for every n∈Z_>0, there exists k≥ 0 such that T^k(n)=1; equivalently, every trajectory eventually reaches the cycle 4→ 2→ 1→ 4.

Syracuse conjecture (Collatz conjecture)

For every positive integer $n\in\mathbb{Z}_{>0}$ , define the Collatz map $T:\mathbb{Z}_{>0}\to\mathbb{Z}_{>0}$ by $T(n)=n/2$ if $n$ is even and $T(n)=3n+1$ if $n$ is odd. The conjecture asserts that for every $n\in\mathbb{Z}_{>0}$ , there exists $k\ge 0$ such that $T^{k}(n)=1$ ; equivalently, every trajectory eventually reaches the cycle $4\to 2\to 1\to 4$ .

References

Primary source

Zenodo (CERN European Organization for Nuclear Research)

Additional references

La conjecture de Syracuse enfin démontrée ? : Étude mathématique détaillée concernant la conjecture de Syracuse via une approche inédite et novatrice — le cadre MSP² — Zenodo (CERN European Organization for Nuclear Research) — Sousa Pereira Mario, Mazzoni Enzo
La conjecture de Syracuse enfin démontrée ? : Étude mathématique détaillée concernant la conjecture de Syracuse via une approche inédite et novatrice — le cadre MSP² — Zenodo (CERN European Organization for Nuclear Research) — Sousa Pereira Mario, Mazzoni Enzo

dynamical-systems number-theory

Posted 3 weeks ago · edited 4 days ago

Progress summary

Refreshed 4 days ago

Claimed solved

A September manuscript claims a complete proof, but the conjecture remains unverified and open.

Attributed to Lothar Collatz in 1937, the conjecture says that every positive integer eventually reaches the cycle $4\to2\to1\to4$ .

Known results

Riho Terras (1976): almost every trajectory eventually falls below its starting value.
Ivan Korec (1994): quantitatively strengthened Terras’s almost-all result.
Terence Tao (2019): almost every trajectory eventually reaches values below any prescribed function tending to infinity.
Computation verifies convergence for starting values below $2^{68}$ , but this is not a proof.

September 2026 claimed proof

On September 7, 2026, Sousa Pereira Mario and Mazzoni Enzo’s manuscript La conjecture de Syracuse enfin démontrée ? claimed a complete proof using the $\mathrm{MSP}^{2}$ framework. No independent verification is reported, so this remains a claim rather than an established solution.

Current status (as of September 2026): Classical partial results and computation are established, but the September claimed proof is unverified and the conjecture remains open.

Sources

Cet article introduit MSP² (écriture regroupée des étapes de Syracuse), une reformulation de la conjecture de Syracuse (Collatz) qui regroupe chaque étape impaire suivie de sa division par 2 inévitable en une seule opération. Cette écriture fait apparaître deux familles d’arbres inverses (racines 6q+1 et 6q−1), dont on établit l’exhaustivité vis-à-vis des vols impairs, ainsi qu’un Tableau Générateur.See full solutionHide full solution

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