The board / additive combinatorics / #1213
Erdős Problem #1213
PROVEDdatabase updated 2026-04-21
The statement as posed is true, and a proof exists.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
A finite sequence `A 0, …, A (s - 1)` has two distinct (nonempty) index intervals `[u, v)` and `[x, y)` with the same sum.
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 \[\sum_{i\in I}a_i=\sum_{j\in J}a_j?\] Hegyvári [He86] has proved the answer is yes, and gives an explicit bound of the shape \(f(a,K) \ll ae^{O(K)}\). Hegyvári believes that the exponential dependence on \(K\) here is not best possible.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved · 1 further variant / known partial result in the Lean file. The authoritative wording is on erdosproblems.com/1213.Prizenone
Topicsadditive combinatorics
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/1213
Attempts
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