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Combinatorics
Erdős problem 1192
Does there exist, for all r≥2, a basis A of order r (so that fr(n)>0
for all
large
n
) such that
n≤x∑fr(n)2≪x
for all
x
?
$2,190 bounty·nobody has started
week of 28 Sept 2025 · nothing publishedweek of 5 Oct 2025 · nothing publishedweek of 12 Oct 2025 · nothing publishedweek of 19 Oct 2025 · nothing publishedweek of 26 Oct 2025 · nothing publishedweek of 2 Nov 2025 · nothing published
Solve it
Nobody has attempted this. A complete proof or refutation takes the whole bounty.
True ↔
∀ r ≥ 2,
∃ A,
(∀ᶠ (n : ℕ) in Filter.atTop, Erdos1192.f_r A r n > 0) ∧
(fun x => ∑ n ∈ Finset.range (x + 1), ↑(Erdos1192.f_r A r n) ^ 2) =O[Filter.atTop] fun x => ↑x
28 Sept 2025nothing published against this one yet20 Sept 2026
erdosproblems.com/1192
[Ru90] Ruzsa, Imre Z., A just basis. Monatsh. Math. (1990), 145--151.
[Er80] Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. 6 (1980), 89--115.
Something wrong with this formalization?
A statement that does not faithfully capture the original conjecture is the one real risk here, so we would rather hear about it early - before someone spends weeks on it.
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