The board / number theory / #318
Erdős Problem #318
SOLVED LEANdatabase updated 2026-09-16
Resolved in some way other than a straight proof or disproof — usual for open-ended questions, or for multi-part problems whose parts went different ways. A solution has also been checked by machine, in Lean.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
A set `A : Set ℕ` is said to have property `P₁` if for any nonconstant sequence `f : A → {-1, 1}`, one can always select a finite, nonempty subset `S ⊆ A \ {0}` such that `∑ n ∈ S, fₙ / n = 0`. This is defined in [Sa82b].
There exists a set `A` with positive density that does not have property `P₁`. #TODO: prove this lemma by assuming `erdos_318.contain_single_even`. The density sits in an existential, so `HasPosDensity` is the *stronger* reading here and weakening it to positive lower density would claim less, which is the opposite of the usual situation for Erdős' "positive density". It also costs nothing: by `erdos_318.variants.contain_single_even` a witness only needs exactly one even element, and the odd numbers together with one even number have density `1 / 2` on the nose.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved · 5 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/318.Prizenone
Topicsnumber theory, unit fractions
Statement in LeanFormal Conjectures
Solution in Leanyes
Original pageerdosproblems.com/318
Attempts
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