587.lean`

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L4DEL: Lean4 Domain Explanation Language

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

2 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 587

Reference: erdosproblems.com/587

Imports (1)

FormalConjecturesUtil

def Erdos587.MaxNotSqSum MaxNotSqSum N is the size of the largest subset A of {1,...,N} such that for all non-empty S ⊆ A, the sum ∑ n ∈ S, n is not a square line 35 source
Source documentation
MaxNotSqSum N is the size of the largest subset A of {1,...,N} such that for all non-empty S ⊆ A, the sum ∑ n ∈ S, n is not a square.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset.sup, Finset, Finset.filter, LE.le, Ne, Bot.bot, Not, IsSquare, Finset.sum, Eq, Finset.powerset, Finset.Icc, OfNat.ofNat, Finset.card
19 instances, coercions and auxiliary constants left out.
```
def MaxNotSqSum (N : ℕ) : ℕ :=
…
```
theorem Erdos587.erdos_587.variants.nguyen_vu Nguyen and Vu proved that $|A| \ll N^{1/3} (\log N)^{O(1)}$ sorry line 43 source
Source documentation
Nguyen and Vu proved that $|A| \ll N^{1/3} (\log N)^{O(1)}$.

depends on sorry what this means
Statement uses: Exists, Nat, And, GT.gt, OfNat.ofNat, Real, Filter.Eventually, LE.le, Erdos587.MaxNotSqSum, HMul.hMul, Real.nthRoot, HPow.hPow, Real.log, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
30 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_587.variants.nguyen_vu : ∃ᵉ (O > 0) (O' > 0),
    ∀ᶠ N in Filter.atTop, (MaxNotSqSum N : ℝ) ≤ O' * Real.nthRoot 3 N * (N : ℝ).log^O := by
…
```
Mentions: Erdos587.MaxNotSqSum, Real.nthRoot