Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 8

Problem 435

claims/: The 1 claim page of Problem 435, one per claimant's result; the problem's standing derives from them.

Statement. Let $n\in\mathbb{N}$ with $n\neq p^k$ for any prime $p$ and $k\geq 0$ . What is the largest integer not of the form

\sum_{1\leq i<n}c_i\binom{n}{i}

where the $c_i\geq 0$ are integers?

Status. Solved; the site labels the problem PROVED (LEAN). The site's commentary credits Hwang and Song with the first proof, and the frontmatter standing is derived from the accepted claim page Hwang and Song's formula, accepted on the site's credit; the formula was found independently in the site's forum by Peake and Cambie, whose posts are recorded on that page and, as forum comments, have no page of their own. The Lean part of the label refers to a formalization of Cambie's forum proof, linked from the same page, which this corpus has not built or audited.

Source. erdosproblems.com/435, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #435, https://www.erdosproblems.com/435.

References.

[HwSo24] W. Hwang and K. Song, The Frobenius problem for Numerical Semigroups generated by binomial coefficients. arXiv:2412.17882 (2024).

Formalization. Statement in formal-conjectures, pinned to the repository's revision of 2026-10-06. Its formal_proof attribute names the Lean development recorded on the claim page.

Progress

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Known Results

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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.

Linked from (9)

Hwang and Song's Frobenius number of the binomial semigroup problems/diophantine_problems/E0435/claims Diophantine Problems and Powers diophantine_problems/hwang_2024_frobenius_problem_numerical_semigroups_generated_binomial Corollary 3.3 (p. 7): Apéry set, Frobenius number, genus and pseudo-Frobenius numbers of the binomial semigroup for n not a prime power Theorem 3.6 (p. 8) and Corollary 3.7 (p. 9): the binomial semigroup for a prime power n = p^m, divided by p Lemma 2.2 (p. 5): C(n, p^k) is congruent to n/p^k modulo n for every prime power p^k dividing n Theorem 0.1 (pp. 2-3) and Remark 0.2: the Apéry set and Frobenius number of the semigroup generated by the binomial coefficients C(n,k)Theorem 3.2 (pp. 6-7): for n not a prime power, C(n,1) and the C(n,p^j) with p^j dividing n minimally generate the binomial semigroup

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