121.lean`

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L4DEL: Lean4 Domain Explanation Language

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

6 declarations, 5 contain sorry, 0 declared as axioms

Erdős Problem 121

References:

erdosproblems.com/121
[Er94b] Erdős, Paul, Some problems in number theory, combinatorics and combinatorial geometry. Math. Pannon. (1994), 261-269.
[Er97] Erdős, Paul, Problems in number theory. New Zealand J. Math. (1997), 155-160.
[Er97e] Erdős, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537.
[Er98] Erdős, Paul, Some of my new and almost new problems and results in combinatorial number theory. Number theory (Eger, 1996) (1998), 169-180.
[ESS95] Erdős, P. and Sárközy, A. and Sós, V. T., On product representations of powers. I. European J. Combin. (1995), 567-588.
[Er38] P. Erdős, On sequences of integers no one of which divides the product of two others and on related problems. Tomsk. Gos. Univ. Ucen Zap. (1938), 74-82.
[Ta24] Tao, T., On product representations of squares. arXiv:2405.11610 (2024).

Imports (1)

FormalConjecturesUtil

def Erdos121.F $F_k(N)$ is the size of the largest $A\subseteq \{1,\ldots,N\}$ such that the product of no $k$ many distinct elements of $A$ is a square line 46 source
Source documentation
$F_k(N)$ is the size of the largest $A\subseteq \{1,\ldots,N\}$ such that the product of no $k$ many distinct elements of $A$ is a square.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: SupSet.sSup, Set.ofPred, Exists, Finset, And, LE.le, Finset.Icc, OfNat.ofNat, Eq, Finset.card, Not, IsSquare, Finset.prod
9 instances, coercions and auxiliary constants left out.
```
noncomputable def F (k N : ℕ) : ℕ :=
…
```
theorem Erdos121.erdos_121 sorry line 68 source
depends on sorry what this means
Statement uses: Iff, False, Asymptotics.IsEquivalent, Nat, Real, Filter.atTop, Erdos121.F, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
18 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos121.erdos_121.variants.odd sorry line 75 source
depends on sorry what this means
Statement uses: Iff, False, Nat, LE.le, OfNat.ofNat, Asymptotics.IsEquivalent, Real, Filter.atTop, Erdos121.F, HAdd.hAdd, HMul.hMul
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos121.erdos_121.variants.tao sorry line 83 source
depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Exists, Real, And, LT.lt, Filter.Eventually, Erdos121.F, HMul.hMul, HSub.hSub, 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 Erdos121.erdos_121.variants.three Erdős, Sós, and Sárközy [ESS95] proved $F_3(N) = (1-o(1))N$ sorry line 89 source
Source documentation
Erdős, Sós, and Sárközy [ESS95] proved $F_3(N) = (1-o(1))N$.

depends on sorry what this means
Statement uses: Asymptotics.IsEquivalent, Nat, Real, Filter.atTop, Erdos121.F, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
18 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_121.variants.three : (fun N : ℕ => (F 3 N : ℝ)) ~[atTop] fun N : ℕ => (N : ℝ) := by
…
```
Mentions: Erdos121.F
theorem Erdos121.erdos_121.variants.two Erdős, Sós, and Sárközy [ESS95] proved $F_2(N)=\left(\frac{6}{\pi^2}+o(1)\right)N$ sorry line 94 source
Source documentation
Erdős, Sós, and Sárközy [ESS95] proved $F_2(N)=\left(\frac{6}{\pi^2}+o(1)\right)N$.

depends on sorry what this means
Statement uses: Asymptotics.IsEquivalent, Nat, Real, Filter.atTop, Erdos121.F, OfNat.ofNat, HMul.hMul, HDiv.hDiv, HPow.hPow, Real.pi
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
42 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_121.variants.two :
    (fun N : ℕ => (F 2 N : ℝ)) ~[atTop] fun N : ℕ => 6 / Real.pi ^ 2 * N := by
…
```
Mentions: Erdos121.F