Problem 246

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claims/: The 7 claim pages of Problem 246, one per claimant's result; the problem's standing derives from them.

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Statement. Let $(a,b)=1$. The set $\{a^kb^l: k,l\geq 0\}$ is complete - that is, every large integer is the sum of distinct integers of the form $a^kb^l$ with $k,l\geq 0$.

Statement (corrected). Let $a,b\geq2$ with $(a,b)=1$. The set $\{a^kb^l: k,l\geq 0\}$ is complete - that is, every large integer is the sum of distinct integers of the form $a^kb^l$ with $k,l\geq 0$.

Notes. The site's wording fixes only $(a,b)=1$. Read as the site words it, it includes $a=1$, where the set $\{b^l\}$ is complete only for $b=2$; for $b=3$ its subset sums, the integers with base-3 digits $0$ and $1$, have density zero, and $a=b=1$ fails as well. The corrected Statement adds $a,b\geq2$, the setting of Birch's theorem [Bi59], of the later literature and of the formal-conjectures statement; the standing judges it.

Status. PROVED (LEAN). The "(LEAN)" suffix is the site's catalog label; the theorem, its acceptance and the Lean development are recorded on the claim pages, and no Lean was built here.

Source. erdosproblems.com/246, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #246, https://www.erdosproblems.com/246.

References.

• [Bi59] Birch, B. J., Note on a problem of Erdős. Proc. Cambridge Philos. Soc. (1959), 370-373.
• [Ca60] Cassels, J. W. S., On the representation of integers as the sums of distinct summands taken from a fixed set. Acta Sci. Math. (Szeged) (1960), 111-124.
• [Er61] Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254.
• [FaCh17] Fang, Jin-Hui and Chen, Yong-Gao, A quantitative form of the Erdős-Birch theorem. Acta Arith. (2017), 301-311.
• [He00b] Hegyvári, N., On the completeness of an exponential type sequence. Acta Math. Hungar. (2000), 127-135.
• [Yu24] Yu, Wang-Xing, On the representation of an exponential type sequence. Publ. Math. Debrecen (2024), 253-261.

Formalization. Statement in formal-conjectures.

Current assessment

The question (site formulation of 2026-09-04). The statement above; PROVED (LEAN), page last edited 7 December 2025. The site's commentary credits the proof to Birch [Bi59] and records that Cassels's more general completeness criterion [Ca60], published the next year, contains it as a consequence.

Claims. The settling result is Birch's theorem (1959), refereed and credited by the site's curator. Cassels's Theorem I, a more general completeness criterion published the next year, gives the statement as an immediate consequence and has its own accepted page, Cassels's criterion. A pending claim, [[problems/diophantine_problems/E0246/claims/2026_09_03_song_yue|Song and Yue's bound $K(p,q)\leq 2p-3$]] (2026-09-03, found with ChatGPT 5.6 Sol), claims a quantitative strengthening of the theorem that bounds the exponent of the second base; it is unreviewed and stays claimed. Hegyvári's explicit bound (2000), Bergelson and Simmons's bound $K(a,b)\leq4a-5$ (Acta Arith. 2017), Fang and Chen's Theorem 1.1 (2017) and Yu's representation by large terms (2024) each prove the statement in a stronger form and have their own accepted pages: Hegyvári 2000, Bergelson and Simmons 2017, Fang and Chen 2017 and Yu 2024.

Quantitative forms. Davenport's remark in [Bi59] is that the exponent $l$ can be kept below a threshold depending only on $a$ and $b$; write $K(a,b)$ for the least $K$ such that the numbers $a^kb^l$ with $0\leq l\leq K$ already form a complete set. The introduction of Fang and Chen's quantitative form (pp. 301--302) records the history of the bounds on $K(a,b)$: Hegyvári [He00b] gave the first explicit bound, quadruply exponential in $a$ and $b$; Fang (2011) and Chen and Fang (2012) improved it to triply exponential; Fang and Chen's own Theorem 1.1 gives a sharper triply exponential $K$, $\log_2\log_2K<b^{2a}$, together with an explicit threshold $B$, $\log_2\log_2\log_2B<b^{2a}$, beyond which every integer is such a sum; and Bergelson and Simmons (2017) proved the linear bound $K(a,b)\leq 4a-5$, by a method that Fang and Chen say seems to give no explicit $B$. The pending claim of Song and Yue asserts $K(a,b)\leq 2a-3$. Yu [Yu24] showed that every large $n$ is a sum of distinct terms all larger than $n/(\log n)^{1+o(1)}$, as the site's commentary records. Of the papers, only [FaCh17] and [Ca60] have library pages, and only [FaCh17] is held.

Formalization and the Lean label. The formal-conjectures statement file, at its commit of 2026-09-18 (ErdosProblems/246.lean), marks erdos_246 solved and names as its formal proof a development in Boris Alexeev's repository, on that repository's main branch rather than at a fixed commit; the development's header declares itself a formalization of Birch's result, written with the system Aristotle, and Birch's claim page links it at a commit of 2026-04-28. Nothing was built, kernel-checked or audited here.

Search scope, 2026-10-07. The site's problem page, commentary, discussion thread and proof-claims page, with the Zenodo record of the pending claim, the Crossref record of [Bi59] and the repository's commit records for dates. arXiv, MathSciNet, zbMATH, Google Scholar and X were not searched.

Remaining gaps. (1) Birch's paper is not held; the claim page cites it by its journal record. (2) The Lean development was not built here, so the claim page lists no formalized evidence. (3) The pending quantitative claim has no review.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• fang_2017_quantitative_form_erdos_birch_theorem
• cassels_1960_representation_integers_as_sums_distinct_summands
• cassels_1960_representation_integers_as_sums_distinct_summands / theorem_i