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

Updated Oct 8

Problem 356

claims/: The 1 claim page of Problem 356, one per claimant's result; the problem's standing derives from them.

Statement. Is there some $c>0$ such that, for all sufficiently large $n$ , there exist integers $a_1<\cdots<a_k\leq n$ such that there are at least $cn^2$ distinct integers of the form $\sum_{u\leq i\leq v}a_i$ ?

Status. PROVED (LEAN). Beker's Theorem 1.2 ([Be23b], Bull. London Math. Soc. 56 (2024), refereed) gives an absolute $c>0$ and, for every $n$ , integers $1\le a_1<\cdots<a_k\le n$ with at least $cn^2$ distinct sums of consecutive terms, so the answer is yes; the site records the solution as Beker's, and the claim page carries the acceptance. Konieczny's theorem ([Ko15]) concerns the permutation variant (Problem 34) and is not a claim on this question. The Lean suffix is the site's label for a 2026 formalization of Beker's result in Boris Alexeev's public repository, registered by the community database; it declares Beker as its informal author, so it is a formalization link on his claim page, not built or audited by this corpus, and gives no formalized evidence.

Source. erdosproblems.com/356, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #356, https://www.erdosproblems.com/356.

References.

[Be23b] Beker, A., On a problem of Erdős and Graham about consecutive sums in strictly increasing sequences. arXiv:2311.10087 (2023); Bull. London Math. Soc. 56 (2024), no. 8, 2749–2759.
[Ko15] Konieczny, J., On consecutive sums in permutations. arXiv:1504.07156 (2015).

Formalization. No statement in formal-conjectures (no 356.lean, and the site lists no formalized statement,). The file src/latest/ErdosProblems/Erdos356.lean of plby/lean-proofs states erdos_356, the problem's existential statement (some $c>0$ works for all large $n$ ), whose proof supplies $c=1/300000$ ; the community database lists the formal status Lean as of its last update (2026-08-24), and the claim page above records the pin.

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Linked library material

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Linked from (13)

Problem 1213 Beker's quadratic consecutive-sum construction problems/integer_sequences/E0356/claims Sequences and Densities of Integers Sequences and Densities of Integers integer_sequences/beker_2023_problem_erdos_graham_about_consecutive_sums Proposition 1.5: an increasing sequence in [n] has at most (c_4 + o(1)) n^2 consecutive sums Theorem 1.2: increasing sequences in [n] with at least c_1 n^2 distinct consecutive sums Theorem 1.3: the random sequence 3i + eps_i has at least c_2 n^2 consecutive sums with positive probability Theorem 1.4: explicit sequences 2i or 2i - 1 with at least c_3 n^2 consecutive sums Theorem 2.1: the partial-sum set of 3i + eps_i has expected additive energy O(n^2)integer_sequences/konieczny_2015_consecutive_sums_permutations number_theory/erdos_1980_old_new_problems_results_combinatorial_number_theory

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