The board / number theory / #357
Erdős Problem #357
OPENdatabase updated 2025-08-31
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
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Let \(f(n)\) be the maximal \(k\) such that there exist integers \(1 \le a_1 < \dotsc < a_k \le n\) such that all sums of the shape \(\sum_{u \le i \le v} a_i\) are distinct.
Let \(f(n)\) be the maximal \(k\) such that there exist integers \(1 \le a_1 < \dotsc < a_k \le n\) such that all sums of the shape \(\sum_{u \le i \le v} a_i\) are distinct. Is \(f(n)=o(n)\)?
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 16 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/357.Prizenone
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/357
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f(n) counts the biggest strictly-increasing sequence up to n where every contiguous run sums to something different. New result: f(n) ≤ n/2 + √n + O(1) — and we didn't just check the final number, we replayed the entire proof by hand and it never once broke.
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