Problem 358

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claims/: The 3 claim pages of Problem 358, one per claimant's result; the problem's standing derives from them.

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Statement. Let $A=\{a_1<\cdots\}$ be an infinite sequence of integers. Let $f(n)$ count the number of solutions to

$$n=\sum_{u\leq i\leq v}a_i.$$

Is there such an $A$ for which $f(n)\to \infty$ as $n\to \infty$? Or even where $f(n)\geq 2$ for all large $n$?

Status. PROVED (LEAN): Tao's 2026 probabilistic construction gives $f(n)\gg\log n$ for all large $n$, answering both questions yes (claim page); the site's label rests on a third-party Lean proof that this corpus has not audited. A thread report of 2026-03-27 found the middle-range step of Proposition 3.1 in the posted manuscript false as written, which Tao said the next revision will correct; no revision had been posted by 2026-10-07, the Lean proof establishes the two answers and not the logarithmic bound, and the claim page records the dispute.

Source. erdosproblems.com/358, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #358, https://www.erdosproblems.com/358.

References.

• [Er77c] Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976) (1977), 43-72.
• [ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).
• [Gu04] Guy, Richard K., Unsolved problems in number theory. Third edition, Problem Books in Mathematics, Springer, New York (2004), xviii+437 pp.; section C2 "Sums of consecutive primes", printed p. 164: Erdős asks for an infinite sequence $1<a_1<a_2<\cdots$ with the number of solutions of $a_i+a_{i+1}+\cdots+a_k=n$ tending to infinity, notes that with $k>i$ it is not even known that $f(n)>0$ for all but finitely many $n$, and that $a_i=i$ gives the number of odd divisors of $n$. Library home: guy_2004_unsolved_problems_number_theory.
• [Mo63] Moser, L., Notes on number theory. III. On the sum of consecutive primes. Canad. Math. Bull. (1963), 159-161.

Formalization. Statement in formal-conjectures.

Current assessment

The site labels the problem PROVED (LEAN) (page last edited 1 April 2026). The problem's standing is solved, proved, through Tao's full claim (claim page), whose evidence is the curator's acceptance: T. F. Bloom records the construction on the problem page as the solution. The Lean file that the site and the formal-conjectures statement file point to, with Codex and GPT-5.6 Sol as its formal authors, proves both answers, $f(n)\to\infty$ and $f(n)\ge2$ for all large $n$, but not the bound $f(n)\gg\log n$; this corpus has not built or audited it, so the claim carries no formalized evidence. A thread report of 2026-03-27 found the middle-range step of Proposition 3.1 false as written; it bears on the logarithmic bound and not on the two answers, Tao said the next revision would correct it, and no revision had been posted by 2026-10-07. No refereed version of the manuscript exists. Two earlier write-ups on the thread fell short: Chojecki's with GPT-5.2 Pro is rejected (claim page) and Sothanaphan's with GPT-5.2 Thinking is withdrawn (claim page).

Search scope, 2026-10-07: the site's problem page and discussion thread, Tao's manuscript of 2026-02-24, Crossref and arXiv for a published or posted version of it (none found).

Known Results

Tao's construction, recorded on the claim page, answers both questions: a set $A$ with $f(n)\gg\log n$ for all large $n$, sharp up to the constant since $\sum_{n\le x}f(n)\le x\log x+O(x)$ for every $A$. Two earlier AI-assisted manuscripts on the thread claimed the result and fell short: Przemek Chojecki's write-up with GPT-5.2 Pro (2026-02-11), rejected (claim page), and Nat Sothanaphan's write-up with GPT-5.2 Thinking (2026-02-18), withdrawn (claim page). The site's remarks record the classical facts: for $A=\mathbb N$ the count $f(n)$ is the number of odd divisors of $n$, so $n$ is a sum of consecutive positive integers if and only if $n$ is not a power of $2$; and for $A$ the primes, the question Erdős and Moser studied [Mo63], even a positive density of represented integers is not known.

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.

• moser_1963_notes_number_theory
• moser_1963_notes_number_theory / equation_5
• moser_1963_notes_number_theory / problems_p161
• guy_2004_unsolved_problems_number_theory
• erdos_1980_noen_mindre_kjente_problemer_i_kombinatorisk
• erdos_1980_noen_mindre_kjente_problemer_i_kombinatorisk / conjecture_p157
• erdos_1980_noen_mindre_kjente_problemer_i_kombinatorisk / question_p157