The board / additive combinatorics / #1192
Erdős Problem #1192
OPENdatabase updated 2026-04-04
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
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For \(A\subset \mathbb{N}\) let \(f_r(n)\) count the number of solutions to \(n=a_1+\cdots+a_r\) with \(a_i\in A\).
Does there exist, for all \(r\geq 2\), a basis \(A\) of order \(r\) (so that \(f_r(n)>0\) for all large \(n\)) such that \[\sum_{n\leq x}f_r(n)^2 \ll x\] for all \(x\)?
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 2 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/1192.Prizenone
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/1192
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Proved the critical-density/lower-energy hierarchy and the exact r=3 block-gluing reduction, and exhaustively certified the minimum cyclic three-basis energy for every modulus 2 through 41; the unreso
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