For every integer m≥ 3, suppose that the positive integers admit a partition Z_>0=B(alpha_1,beta_1)sqcup…sqcup B(alpha_m,beta_m) into Beatty sequences B(alpha_i,beta_i)=lfloor n alpha_i+beta_i rfloor:n∈Z_>0 with distinct moduli alpha_i. Writing delta_i=1/alpha_i for the density of the ith sequence, one must have delta_1,…,delta_m=frac12^m-1,frac22^m-1,frac42^m-1,…,frac2^m-12^m-1.

Fraenkel’s conjecture

For every integer $m\ge 3$ , suppose that the positive integers admit a partition $\mathbb{Z}_{>0}=B(\alpha_1,\beta_1)\sqcup\cdots\sqcup B(\alpha_m,\beta_m)$ into Beatty sequences $B(\alpha_i,\beta_i)=\{\lfloor n\alpha_i+\beta_i\rfloor:n\in\mathbb{Z}_{>0}\}$ with distinct moduli $\alpha_i$ . Writing $\delta_i=1/\alpha_i$ for the density of the $i$ th sequence, one must have $\{\delta_1,\ldots,\delta_m\}=\left\{\frac{1}{2^m-1},\frac{2}{2^m-1},\frac{4}{2^m-1},\ldots,\frac{2^{m-1}}{2^m-1}\right\}$ .

References

Primary source

arXiv

Additional references

A Proof of Fraenkel's Conjecture — arXiv

Posted 1 month ago

Progress summary

Refreshed 4 weeks ago

Claimed solved

A September 2026 preprint claims a complete proof of Fraenkel’s conjecture, but independent verification is not yet available.

Fraenkel’s conjecture predicts a unique set of densities for partitions of the positive integers into at least three Beatty sequences with distinct moduli. The binary pattern was identified by Fraenkel and the stronger uniqueness formulation was recorded by Erdős and Graham.

Known results

$m=3$ : Morikawa.
$m=4$ : Altman, Gaujal, and Hordijk.
$m=5,6$ : Tijdeman.
$m=7$ : Barát and Varjú; the general case $m\ge 8$ was previously open.

September 2026 claimed proof

On September 1, 2026, Hu Tan and Ying Zhang’s preprint A Proof of Fraenkel’s Conjecture claims the result for every $m\ge 3$ . Its central step is a one-third theorem, followed by induction; the authors also report assistance from GPT-5.6 Sol, OpenAI Codex, and Ziv. The proof, finite checks, and planned Lean formalisation remain independently unverified.

Current status (as of September 2026): Cases through $m=7$ are established, while the full conjecture is only claimed in an unrefereed preprint and remains unverified.

Sources

No solutions have been posted yet.