764.lean`

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

3 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 764

References:

erdosproblems.com/764
[Er65b] Erdős, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244.
[Er70c] Erdős, P., Some problems in additive number theory. Amer. Math. Monthly (1970), 619-621.
[Va72] Vaughan, R. C., On the addition of sequences of integers. J. Number Theory (1972), 1-16.

Imports (1)

FormalConjecturesUtil

def Erdos764.tripleRep The number of representations of n as an ordered sum of three elements of A, $1_A\ast 1_A\ast 1_A(n)$ line 41 source
Source documentation
The number of representations of n as an ordered sum of three elements of A, $1_A\ast 1_A\ast 1_A(n)$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Nat
Proof uses: AdditiveCombinatorics.sumConv, Set.indicatorOne
4 instances, coercions and auxiliary constants left out.
```
noncomputable def tripleRep (A : Set ℕ) : ℕ → ℕ := 𝟙_A ∗ 𝟙_A ∗ 𝟙_A
…
```
theorem Erdos764.erdos_764 sorry line 56 source
depends on sorry what this means
Statement uses: Iff, False, Exists, Set, Nat, Real, And, LT.lt, OfNat.ofNat, Asymptotics.IsBigO, Filter.atTop, HSub.hSub, Finset.sum, Finset.range, HAdd.hAdd, Erdos764.tripleRep, HMul.hMul
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
34 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos764.erdos_764.variants.vaughan Vaughan [Va72] showed that $\sum_{n\leq N} 1_A\ast 1_A\ast 1_A(n) = cN+o\left(\frac{N^{1/4}}{(\log N)^{1/2}}\right)$ is impossible sorry line 67 source
Source documentation
Vaughan [Va72] showed that $\sum_{n\leq N} 1_A\ast 1_A\ast 1_A(n) = cN+o\left(\frac{N^{1/4}}{(\log N)^{1/2}}\right)$ is impossible.

depends on sorry what this means
Statement uses: Not, Exists, Set, Nat, Real, And, LT.lt, OfNat.ofNat, Asymptotics.IsLittleO, Filter.atTop, HSub.hSub, Finset.sum, Finset.range, HAdd.hAdd, Erdos764.tripleRep, HMul.hMul, HDiv.hDiv, HPow.hPow, Real.sqrt, Real.log
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
50 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_764.variants.vaughan : ¬ ∃ (A : Set ℕ) (c : ℝ), 0 < c ∧
    (fun N : ℕ ↦ (∑ n ∈ Finset.range (N + 1), tripleRep A n : ℝ) - c * N) =o[atTop]
      fun N ↦ (N : ℝ) ^ (1 / 4 : ℝ) / √(Real.log N) := by
…
```
Mentions: c, Erdos764.tripleRep