Problem 354

Let $\alpha,\beta\in \mathbb{R}_{>0}$ such that $\alpha/\beta$ is irrational. Is the multiset

\{ \lfloor \alpha\rfloor,\lfloor 2\alpha\rfloor,\lfloor 4\alpha\rfloor,\ldots\}\cup \{ \lfloor \beta\rfloor,\lfloor 2\beta\rfloor,\lfloor 4\beta\rfloor,\ldots\}

complete? That is, can all sufficiently large natural numbers $n$ be written as

n=\sum_{s\in S}\lfloor 2^s\alpha\rfloor+\sum_{t\in T}\lfloor 2^t\beta\rfloor

for some finite $S,T\subset \mathbb{N}$ ?

What if $2$ is replaced by some $\gamma\in(1,2)$ ?

Source: erdosproblems.com/354

Number theory
Complete sequences

Status

Open

No claim settles this problem.

Open, the site's label OPEN, which attaches to the pair of questions. The first question (base $2$ ) has two accepted partial claims: the bounty site Conjectures.io's certified Lean proof, answering it yes for all $\alpha,\beta>0$ with $\alpha/\beta$ irrational, and Hegyvári's 1989 theorem (Acta Math. Hungar.; refereed), answering it yes when one coefficient is a dyadic rational and the other is not. It also has one pending partial claim, the Yu–Chen manuscript, claiming the stronger strong completeness. The second question has a pending partial claim under each of its two readings: Kitamura's Lean proof at one base answers yes under the reading "for some $\gamma\in(1,2)$ ", and Geneson's Salem-base counterexample answers no under the reading "for every $\gamma\in(1,2)$ ". No claim settles the pair of questions, so the derived standing is open; the full answer to the first question rests on a bounty site's acceptance alone, with no refereed publication and no erdosproblems.com acceptance, and the Current assessment records its provenance and limits.

ErGr80: Erdős, P. and Graham, R. L., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathématique 28 (1980), printed p. 58, the site's source key. Library home: Erdős and Graham 1980.
FaHe25: Fang, J.-H. and He, J.-Y., On a problem of Erdős and Graham. Acta Math. Hungar. 175 (2025), 532--542. Not held; site-attested.
Gr71: Graham, R. L., On sums of integers taken from a fixed sequence. Proceedings of the Washington State University Conference on Number Theory (1971), 22--40; Question 12, printed p. 36, is the problem in this wording, both questions included. Library home: Graham 1971; result page Question 12.
He89: Hegyvári, N., Some remarks on a problem of Erdős and Graham. Acta Math. Hungar. 53 (1989), 149--154, DOI 10.1007/BF02170065. Not held; site-attested, and as summarized by [Fa26], [Ge26] and [YuCh26]; claim page Hegyvári 1989.
He91: Hegyvári, N., On complete sequences. Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 34 (1991), 7--10. Library home: Hegyvári 1991.
He94: Hegyvári, N., On sumset of certain sets. Publ. Math. Debrecen 45 (1994), 115--122. Library home: Hegyvári 1994.
JiMa24: Jiang, X.-W. and Ma, W.-X., A conjecture of Hegyvári. Int. J. Number Theory 20 (2024), 915--933. Not held; site-attested.
JW26: JenW1N (solver handle), Lean proof of Erdős problem 354, part (i). Conjectures.io record 815c1d5f-3afb-4430-8e2c-260d9038f5b0; verified, review approved 15 September 2026, certified 16 September 2026. Library home: JenW1N 2026; result page JenW1N 2026, target.
YuCh26: Yu, Y. and Chen, K., Erdős Problem 354(i): Strong Completeness of Two Dyadic Floor Sequences. Manuscript dated 13 September 2026, 17 pp., in the GitHub repository Andrewyzzz/erdos354 (proof/FORMALIZED_PROOF.pdf at the revision of 20 September 2026, the head on 2026-09-28, pinned on the claim page); Theorem, p. 1; unrefereed. Library home: Yu and Chen 2026; result page Theorem.
Ki26: Kitamura, K., A Lean proof of Erdős Problem 354(ii). GitHub repository KitaKen1/erdos-354-part-ii, revision of 5 September 2026 (pinned on the claim page); theorem erdos_354_part_ii_solved; unreviewed. Library home: Kitamura 2026; result page Kitamura 2026, erdos_354_part_ii_solved.
Ge26: Geneson, J., Deletion thresholds and exponential examples for complete sequences. arXiv:2609.25107v1 [math.CO] (20 September 2026), 14 pages; Corollary 12, p. 12; unrefereed. Library home: Geneson 2026; result page Corollary 12.
Fa26: Fan, S., Strongly complete sets and a conjecture of Erdős. arXiv:2607.14071 (v1 15 July 2026; v5 16 September 2026, the version cited; v4 of 9 September 2026, the version the bounty site's review cites); Corollary 1.2 (p. 4) and Remark 4.2 (v5, p. 20; Remark 4.1 of v4, p. 19); unrefereed. Library home: Fan 2026; result page Remark 4.2.

Problem 354

Status

Proof claims

Discussion