Problem 1213

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claims/: The 1 claim page of Problem 1213, one per claimant's result; the problem's standing derives from them.

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Statement. Let $a,K\geq 1$. Does there exist $f(a,K)$ such that if

$$a=a_1<\cdots <a_s$$

is a sequence of integers with $a_s> f(a,K)$ and with bounded gaps $a_{i+1}-a_i\leq K$ then there are two distinct intervals $I$ and $J$ such that

$$\sum_{i\in I}a_i=\sum_{j\in J}a_j?$$

Formulation. The Statement is the site's wording(page last edited 10 April 2026). An interval is a nonempty set of consecutive indices $\{u,\ldots,v\}\subseteq\{1,\ldots,s\}$, and the sums run over the terms $a_i$ with those indices; the two intervals are distinct and may overlap (two distinct intervals with equal sums are never nested, since the terms are positive; an observation made here). The sequence starts at the prescribed value $a$, its consecutive gaps are at most $K$, and the question is whether a bound $f(a,K)$ on the last term forces two intervals with the same sum. The site's commentary records the answer as yes with a bound of the shape $f(a,K)\ll ae^{O(K)}$. The site's only source key is [He86]; a separate thanks line follows the two-sentence commentary.

Status. Proved, the site's label. The status-defining source is Hegyvári's paper On consecutive sums in sequences (Acta Math. Hungar. 48 (1986), no. 1--2, 193--200, refereed), which the site credits with the affirmative answer and an explicit bound of the shape $f(a,K)\ll ae^{O(K)}$; the site adds that the author thinks the exponential dependence on $K$ can be improved. The claim is recorded on its claim page and accepted on the refereed publication and the site's credit. Theorem 3 (p. 197) of the paper states $f(a,K)<(a+K/2)e^{K+1}+Ke^{2K+2}$, where $f(a,K)$ is "the largest integer with the following property: There exists an increasing sequence $a=a_1<a_2<\ldots<a_s=f(a,K)$ such that $a_{i+1}-a_i\le K$, $i=1,2,\ldots,s-1$ and all $c$-sums are different". Two equal $c$-sums are two distinct intervals of equal sum, overlapping intervals included, so this is the question's $f(a,K)$ with explicit constants.

Source. erdosproblems.com/1213, accessed 2026-09-18. Cite as: T. F. Bloom, Erdős Problem #1213, https://www.erdosproblems.com/1213, accessed 2026-09-18.

References.

• [He86] Hegyvári, N., On consecutive sums in sequences. Acta Math. Hungar. 48 (1986), no. 1--2, 193--200, DOI 10.1007/BF01949064 (received October 2, 1984). The site's only source. Section 3 holds the question, the definition of $f(a,K)$, the small values, the $K=1$ estimate, Theorem 3 (p. 197) and its proof (pp. 197--198). Library home: hegyvari_1986_consecutive_sums_sequences; result page theorem_3.
• [Be23] Beker, A., On a problem of Erdős and Graham about consecutive sums in strictly increasing sequences. arXiv:2311.10087v1 (16 November 2023), 9 pp.; its p. 1 cites [He86] for a different theorem (below). Library home: beker_2023_problem_erdos_graham_about_consecutive_sums (a card of Problem 356; it restates no result on this problem).

Formalization. Statement only. The catalog google-deepmind/formal-conjectures holds ErdosProblems/1213.lean, added on 2026-09-20 (the linked revision, the file's only change), stating the question with answer(True) and a sorry body and pointing its formal_proof attribute at a sorry-free Lean development in Boris Alexeev's lean-proofs repository that declares itself a formalization of Hegyvári's solution but proves its own explicit bound; the community database (teorth/erdosproblems, copy of 2026-10-06) records a formalized statement and formal status unformalized. The development is linked on Hegyvári's claim page; nothing was built or audited here.

Current assessment

The question (site formulation of 2026-09-18). The statement above; labeled proved with an affirmative answer, last edited 10 April 2026. The commentary consists of two sentences: that Hegyvári [He86] proved the answer yes with an explicit bound of the shape $f(a,K)\ll ae^{O(K)}$, and that he believes the exponential dependence on $K$ is not the truth. The discussion thread and the proof-claim tab were empty on 2026-09-18; the community database (copy of 2026-10-06) records the problem proved and a formalized statement, with formal status unformalized (see Formalization above).

The status-defining source. [He86] is a refereed paper in Acta Mathematica Hungarica (volume 48, issue 1--2, pp. 193--200, received October 2, 1984); its library card has the result page Theorem 3. Section 3 (p. 197) opens with the question, attributed to Erdős by personal communication: "Is it true that if $\{a_i\}$ is an increasing sequence and $a_{i+1}-a_i\le K$, $K\in\mathbf N$, then there exist at least two $c$-sums which are equal if $a_i$ is large enough? The answer is yes and we establish this statement in a quantitative form." A $c$-sum is a sum $\sum_{u\le i\le v}a_i$ over an index pair $1\le u\le v\le s$ (p. 193), so "all $c$-sums are different" excludes two distinct intervals of equal sum, overlapping or not, exactly as the statement above; the sequence starts at $a_1=a$; and $f(a,K)$ is defined as the largest last term of such a sequence, the same quantity as the question's threshold. Theorem 3: $f(a,K)<(a+K/2)e^{K+1}+Ke^{2K+2}$. Since $(a+K/2)e^{K+1}+Ke^{2K+2}\le a(e^{K+1}+2Ke^{2K+2})$ for $a\ge1$, this is the site's $f(a,K)\ll ae^{O(K)}$ with explicit constants. The proof is a one-page count: the blocks $a_{i+1}+\cdots+a_{i+j}$ with $c$-sum below $D$ number more than $S'=\frac DK\log A-\frac{A(a+K/2)}K-\frac{(A+2)^2}4$ over lengths $j\le A$ (the paper's (3.6)), and the paper asserts (its (3.7)) that for $A=[e^{K+1}]$ this exceeds $D$ once $D>L=(a+K/2)e^{K+1}+Ke^{2K+2}$, so that two of them share a value. The same page prints $f(1,1)=2$, $f(2,1)=4$, $f(1,2)=7$, $f(2,2)=10$ and, from Theorem 2 on translates of $\{1,\ldots,k\}$, $a+(1+o(1))2\sqrt a<f(a,1)<a+(1+o(1))5\sqrt a$ for $a>a_0$: the paper's only lower bounds. Three observations on the printed text, not review verdicts: the theorem prints the strict inequality while the proof's last line concludes $f(a,K)\le L$; the paper prints no remark on whether the exponential dependence on $K$ is best possible, so the site's sentence about the author's belief has no printed counterpart in it; and the printed step from (3.6) to (3.7) fails for large $a$ at every $K$, since $S'\ge D$ needs $D(\log A-K)>A(a+K/2)+K(A+2)^2/4$, and with $A=[e^{K+1}]$ the coefficient $A/(\log A-K)$ of $a$ exceeds $e^{K+1}$ (for $K=1$, $A=7$, it is $7.4003$ against $e^2\approx7.3891$): for $K=1$, $a=10^4$ and $D=L+1$ one finds $S'-D\approx-74$, and the step first fails near $a=3000$. The conclusion survives through the slack discarded in (3.6): the floor sum $S$ of (3.5) keeps the harmonic sum $\sum_{j\le A}1/j\ge\log A+\tfrac12$ that (3.6) replaces by $\log A$, and $S\ge D$ at $D=\lfloor L\rfloor+1$ at the same points ($S-D=47770$ for $K=1$, $a=10^4$; $S-D>6\cdot10^9$ for $K=2$ and $K=3$ at $a=10^9$). Attestation beyond the site: the paper [Be23] cites [He86] on p. 1 for another of its results, that for the non-monotone form of Erdős and Graham's consecutive-sums question "an affirmative answer was given by Hegyvári, who showed more strongly in [5] that one can find a sequence of length $\left(\frac{1}{3}+o(1)\right)n$ in $[n]$ with all consecutive sums distinct" (the paper's Theorem 1); this is not a restatement of the bounded-gap theorem, so [Be23] bears on another question, not on this one. The label PROVED crediting [He86], given by a curator independent of the author, and the refereed venue are the documented acceptance; the community database records the problem proved; no dispute was found.

Search scope. None of the routes below found the paper's text, a restatement of the theorem, an improvement of the $e^{O(K)}$ dependence, or a dispute.

• The site: problem page, discussion thread and proof-claim tab as of 2026-09-18; the formal-conjectures catalog, which held no file for this problem before 2026-09-20 (the statement file added that day is recorded under Formalization above); the community database as of 2026-09-18.
• Crossref: the record of [He86] (bibliographic query).
• arXiv: the API queries abs:"bounded gaps" AND abs:"consecutive sums" and all:Hegyvári AND abs:"consecutive sums" (no records); the abstract search for "Erdős problem 1213" (no record).
• The paper [Be23], at p. 1 and its reference list (p. 9).

Not searched: MathSciNet, zbMATH, Google Scholar, X.

Remaining gaps. (1) Proof: no review of Theorem 3's proof beyond the journal's refereeing is recorded, and the printed step from (3.6) to (3.7) needs the repair described above. (2) Lower bounds: the paper gives only the four small values and the $K=1$ estimate, and no source found bounds $f(a,K)$ from below for $K\ge2$. (3) Whether the dependence on $K$ can be improved below exponential is open; the site attributes to the author the belief that it can, and the paper prints no such remark.

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.

• hegyvari_1986_consecutive_sums_sequences
• hegyvari_1986_consecutive_sums_sequences / theorem_3